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NAVIGATION 


AND 


NAUTICAL   ASTRONOMY 


BY 

PROF.  J.  H.  C.  COFFIN 

Lai e  Professor  of  Astronomy,  Navigation,  and  Surveying  at  the 
U.S.  Naval  Academy 


REVISED 

BY 

COMMANDER  CHARLES  BELKNAP 

U.    S.  NAVY 

PREPARED 

FOR   THE    USE   OF  THE   U.S.  NAVAL  ACADEMY 


Seventh  Edition,  Revised 


NEW    YORK 
D.     VAN     NOSTRAND     COMPANY 

1898 


Copyright,  1898, 
By  D.  Van  Nostrand  Company. 


Typography  by  C.  J.  Peters  <fe  Son, 
Boston. 


Plimpton  ^resa 

H.  M.  PLIMPTON  &  CO.,  PRINTERS  A  BINDERS, 
NORWOOD,  MASS.,  U.S.A. 


PUBLISHER'S   NOTE. 


The  continued  demand  for  the  late  Professor  Coffin's 
treatise,  at  the  Naval  Academy  and  by  the  profession,  ren- 
dered necessary  a  thorough  revision,  which  has  been  made 
by  Commander  Charles  Belknap,  U.S.N.,  who  has  brought 
the  work  fully  up  to  date,  all  the  examples  being  based  on 
the  Ephemeris  of  1898. 

Commander  Belknap  being  called  to  Manila,  was  unable 
to  see  the  work  through  the  press,  and  in  his  absence  the 
proofs  were  read  by  Lieutenant  E.  H.  Tillman  U.S.K,  As- 
sistant Instructor  in  Navigation,  U.  S.  Naval  Academy,  to 
whom  the  publishers  take  this  means  of  expressing  their 
acknowledgment. 

October,  1898. 

iii 


M347103 


CONTENTS. 


CHAPTER. 

L  The  Sailings  :  Plane,  pp.  1-4  ;  Traverse,  pp.  5-6  ;  Parallel,  pp.  7-8  ; 
Middle  Latitude,  pp.  8-10  ;  Mercator's,  pp.  11-20  ;  Mercator  Chart, 
pp.  20-24  ;  Great-Circle  Sailing,  pp.  24-31. 

II.  Refraction,  pp.  32-36. 

Dip  of  the  Horizon,  pp.  36-41.    Distance  of  the  horizon,  p.  63. 
Parallax,  pp.  41-44  ;  Apparent  Semi-Diameters,  pp.  45^7 ;  Aug- 
mentation of  the  Moon's  Semi-Diameter,  p.  47. 

III.  Time  :  Sidereal,  Solar,  Apparent,  Mean,  pp.  48-50  ;  Equation  of  Time, 

p.  50  ;  Astronomical,  Civil,  and  Sea  Time,  pp.  49-51  ;  The  Rela- 
tion of  Local  Times  and  Longitudes,  pp.  52-54  ;  Standard  Time, 
p.  56. 

IV.  The  Nautical  Almanac,  pp.  57-73  ;  Interpolation,  pp.  57-60  ;   To 

find  a  required  Quantity  for  a  given  Mean  Time,  pp.  61-63  :  Sun's 
Right  Ascension,  etc.,  for  apparent  Time,  pp.  66-68  ;  Mean  Time 
of  the  Moon's  Transit,  pp.  68-69  ;  Of  a  Planet's  Transit,  pp.  69-70  ; 
Right  Ascension  and  Declination  of  the  Moon  or  a  Planet  at  the 
Time  of  Transit,  p.  70  ;  Greenwich  Mean  Time  of  a  Lunar  Dis- 
tance, pp.  71-74. 

V.  Conversion  of  the  several  kinds  of  Time  —  Relation  of  Time  and 
hour-angle.  —  To  convert  :  —  Apparent  into  Mean  Time  or  Mean 
into  Apparent,  p.  75  ;  Mean  into  Sidereal  Time  or  Sidereal  into 
Mean,  p.  76  ;  Mean  Time  at  a  given  place  into  Sidereal,  pp.  77-79  ; 
Examples  of  conversion  of  Time,  p.  80  ;  Sidereal  Time  at  any 
place  into  Mean,  pp.  81-83  ;  Examples  of  Conversion  of  Time,  pp, 
83-84.  Relation  of  Hour-Angle  and  Time,  pp.  84-89  ;  Examples, 
89-90. 

VI.   The  Astronomical  Triangle,  p.  91-92. 

Altitude  and  Azimuth  for  a  Given  Time,  pp.  95-98  ;  Azimuth  from 
an  Observed  Altitude,  pp.  99-100  ;  Amplitude,  p.  101. 

Examples,  pp.  102-104  ;  Hour-Angle  and  Local  Time  from  an 
Observed  Altitude,  pp.  105-109  ;  Hour-Angle  on,  or  nearest  to,  the 
Prime  Vertical,  pp.  112-113  ;  Examples,  110-113. 


vi  CONTENTS. 

Chapter. 
VII.  Latitude  from  a  Meridian  Altitude,  pp.  114-115  ;  Examples,  pp. 
11(5-117;  An  Altitude  at  any  Time,  pp.  121-122;  An  Altitude 
near  the  Meridian,  pp.  125-127;  Examples,  pp.  128-130  ;  Circum- 
Meridian  Altitudes,  pp.  130-133  ;  Example,  pp.  133-134.  To  find 
the  latitude  from  two  altitudes  when  the  time  is  not  known, 
Chauvenet's  Method,  pp.  134-136  ;  Example,  p.  137  ;  An  Altitude 
of  Polaris,  pp,  138-140. 

VIII.  Chronometers,  p.  142  ;  To  find  the  Rate  of  a  Chronometer,  p.  145  ; 
Comparison  of  Watches  and  Chronometers,  p.  146  ;  To  find  the 
Chronometer  Correction  from  Single  Altitudes,  p.  147  ;  Double 
Altitudes,  p.  149  ;  Equal  Altitudes  of  a  Star,  p.  153  ;  Equal  Alti- 
tudes of  the  Sun,  p.  154  ;  Equal  Altitudes  of  the  Moon,  or  a 
Planet,  p.  159  ;  By  Meridian  Transits,  p.  164  ;  By  time  signals, 
p.  165. 
Longitude,  p.  166  ;  By  a  Portable  Chronometer,  p.  167  ;  Reports  of 
Longitudes,  p.  169  ;  Finding  the  Local  Time  from  Single  Alti- 
tudes, p.  169  ;  Double  Altitudes,  p.  173  ;  Two  Altitudes  of  the 
Sun  near  Noon  (Littrow's  Method),  p.  175  ;  Equal  Altitudes,  p. 
178  ;  From  Meridian  Transits,  p.  183  ;  Longitude  from  Lunar 
Distances,  pp.  184-192  ;  Other  Lunar  Methods,  p.  192. 

IX.  Finding  a  Line  of  Position  (Sumner's  Method),  p.  194. 

Latitude  and  Longitude  by  Two  Lines  of  Position,  p.  200  ;  Redu- 
cing an  Altitude  for  Change  of  Position,  p.  207  ;  Other  Methods, 
p.  210.  To  find  the  latitude  by  rate  of  change  of  altitude  near 
the  prime  vertical  (Prestel's  Method),  p.  211. 

X.  Azimuth  of  a  Terrestrial  Object,  p.  213  ;  Azimuth  of  a  Terres- 
trial Object,  by  a  Theodolite  or  other  Azimuth  Instrument,  pp. 
214-215  ;  By  a  Sextant  (sometimes  called  an  Astronomical  Bear- 
ing), p.  217-221. 


NAVIGATION. 


CHAPTER   I. 

THE     SAILINGS. 

PLANE     SAILING. 

1.  Suppose  the  compass-needle  constantly  to  point  to  the 
north,  a  ship  which  is  steered  by  it  upon  any  given  course 
must  cross  every  meridian  at  the  same  angle/  namely,  the 
angle  given  by  the  compass.  She  does  not  sail  on  a  great 
circle,  except  when  she  sails  on  the  equator,  east  or  west,  or 
on  a  meridian,  north  or  south.  All  other  great  circles  inter- 
sect successive  meridians  at  varying  angles. 

A  line  which  makes  the  same  angle  with  each  successive 
meridian  is  called  a  loxodromic  curve ;  in  old  nautical  works, 
a  rhumb-line  /  more  commonly,  the  ship's  track. 

The  constant  angle  which  it  makes  with  the  meridian  is 
the  course,  and  is  called  the  true  course,  to  distinguish  it  from 
the  compass  course. 

The  length  of  the  line  considered,  or  the  distance  sailed, 
is  called  the  distance. 

The  corresponding  increase  or  decrease  of  latitude  is  the 
difference  of  latitude. 

The  distance  between  the  meridian  left,  and  that  arrived 
at,  measured  on  a  parallel  of  latitude,  is  the  departure  on  that 
parallel. 

1 


Z  NAVIGATION. 

The  distance  between  these  meridians,  measured  on  the 
equator,  is  the  corresponding  difference  of  longitude. 


2.  The  following  notation  will  be  employed  ;  the  refer- 
ences being  to  Fig.  1,  in  which  C  A  represents  a  portion  of 
a  loxodromic  curve  : 

O  =  B  C  A,  the  course. 

d  =  CA,  the  distance. 
?=CB=:EA,  the  difference  of 
latitude. 

p  =  the  departure  :  C  E  in  the 
latitude  of  C,  BA  in  the 
latitude  of  B,  FG  in  the 
latitude  of  F. 

D  =  C'A',  the  difference  of  longi- 
tude. 

Z  =  C  C,  the  latitude  left,  )  +  when  North. 

U  =  A'A,  the  latitude  arrived  at,  )  —  when  South. 

X  =  the  longitude  left,  )  -f-  when  West. 

A'  =  the  longitude  arrived  at,         )  —  when  East. 

Evidently  I  =  U —  L,         D  =  V  -  X ;   ) 

whence  1! =  L  +  I,  X'=  A  +  D,  i 

in  which  attention  must  be  paid  to  the  signs,  or  names.* 


Fig.  I. 


(i) 


3.  If  the  distance  is  very  small,  so  that  the  curvature  of 
the  earth  may  be  neglected,  then  C  A  may  be  regarded  as  a 
right  line,  and  the  triangle  CAB  as  a  right  plane  triangle. 
From  this  we  have 

*  If  IT.  and  W.  are  regarded  as  positive,  S.  and  E.  are  negative,  and 
may  be  treated  as  such,  without  the  formality  of  substituting  the  signs 
+  and  — . 


PLANE   SAILING. 


or 


cos  6  =  — , 
a 


sinff=£, 
d 


tan  C=^; 


/  =  c?  cos  C,        p  —  d  sin  C,       p  =  I  tan  (7, 


(2) 
(3) 


in  which p  is  the  departure  in  the  latitude  of  C  or  A;  indif- 
ferently, as  their  distance  is  very  small. 

The  Traverse  Table,  or  Table  of  Right  Triangles,  contains 
I  and  p  for  different  values  of  C  and  d.     Table  1  in  "  Bow- 
ditch's  Navigator  "  contains  I  and  p  for  each 
unit  of  d  from  1  to  300,  and  for  each  quarter- 
point  of  C.    Table  2  contains  them  for  each 
unit  of  d  and  each  degree  of  C. 

These  quantities  form  a  plane  right  tri- 
angle (Fig.  2),  in  which 


d  is  the  hypothenuse, 

C  one  of  the  angles, 

I  the  side  adjacent  } 

p  the  side  opposite  ) 


that  angle. 


Fig.  2. 


In  the  Tables,  the  columns  of  distance,  difference  of  lati- 
tude, and  departure  might  be  appropriately  headed,  respect- 
ively, hypothenuse,  side  adjacent,  and  side  opposite. 


4.  The  first  two  of  equations  (3)  afford  the  solution  of  the 
most  common  elementary  problem  of  navigation  and  survey- 
ing, viz. : 

Problem  1.  Given  the  course  and  distance,  to  find 
the  difference  of  latitude  and  departure,  the  distance 
being  so  small  that  the  curvature  of  the  earth  may  be 
neglected. 

These  equations  also  afford  solutions  of  all  the  cases  of 
Plane  Sailing.     (Bowd.,  Art.  113.) 


4  NAVIGATION. 

5.  Problem  2.  Given  the  course  and  distance,  to  find 
the  difference  of  latitude  and  departure,  when  the  dis- 
tance is  so  great  that  the  curvature  of  the  earth  cannot 
be  neglected. 

Solution.    Let  the  distance  C  A  (Fig.  1)   be  divided  into 
parts,  each  so  small  that  the  curvature  of  the  earth  may  be 
neglected  in  computing  its  corresponding  difference  of  lati- 
tude and  departure.     For  each  such  small  distance,  as  c  a, 
I  =  d  cos  Cy  p  =  d  sin  C. 

Representing  the  several  partial  distances  by  dlf  d2,  d3, 
etc.,  the  corresponding  values  of  I  and  p  by  l1}  l2,  ls,  etc.,  and 
ply  p2,  p3,  etc.,  and  the  sums  respectively  by  [«?],  [/],  [jo],  we 

have 

h  +  h  +  h  +  etc-  =  (di  -\-  d2  -\-  d3,  etc.)  cos  C, 

^1+^2+^3  +  etc.  =  (di  +  d2  +  ds,  etc.)  sin  67; 
or,  [  £]  =  [d~\  cos  C,  \_p~]  =  [d~\  sin  C. 

Since  the  distance  between  two  parallels  of  latitude  is  the 
same  on  all  meridians,  the  sum  of  the  several  partial  differ- 
ences of  latitude  will  be  the  whole  difference  of  latitude ;  As 
in  Fig.  1. 

CB  =  EA  =  the  sum  of  all  the  sides,  c  b,  of  the 
small  triangles ; 

and  we  shall  have  generally,  as  in  Problem  1, 

I  =  d  cos  C. 

We  also  have  7    .     „ 

p  =  d  sin  C, 

if  we  regard  p  as  the  sura  of  the  partial  departures,  each  being 
taken  in  the  latitude  of  its  triangle  ;  so  that  the  difference  of 
latitude  and  departure  are  calculated  by  the  same  formulas, 
when  the  curvature  of  the  earth  is  taken  into  account,  as 
when  the  distance  is  so  small  that  the  curvature  may  be  dis- 
regarded j  or,  in  other  words,  as  if  the  earth  were  a  plane. 


TRAVERSE  SAILING.  5 

But  the  sum  of  these  partial  departures,  b  a  of  Fig.  1,  is 
evidently  less  than  C  E,  the  distance  between  the  meridians 
left  and  arrived  at  on  the  parallel  C  E,  which  is  nearest  the 
equator ;  and  greater  than  B  A,  the  distance  of  these  merid- 
ians on  the  parallel  B  A,  which  is  farthest  from  the  equator. 
But  it  is  nearly  equal  to  F  G,  the  distance  of  these  meridians 
on  a  middle  parallel  between  C  and  A ;  and  we  take  then 
X0  =  \  {L  -\-  IJ),  as  the  latitude  for  the  departure,  p. 

6.  Middle  Latitude  Sailing  regards  the  departure,  p,  as 
the  distance  between  the  meridian  left  and  that  arrived  at  on 
the  middle  parallel  of  latitude  ;  or  takes  X0  =  J  (X  -f-  L'). 

TRAVERSE     SAILING. 

7.  If  the  ship  sail  on  several  courses,  instead  of  a  single 
course,  she  describes  an  irregular  track,  which  is  called  a 
Traverse. 

Problem  3.  To  reduce  several  courses  and  distances 
to  a  single  course  and  distance,  and  find  the  correspond- 
ing1 differences  of  latitude  and  departure. 

Solution.  If  in  Fig.  1  we  regard  C  as  different  for  each 
partial  triangle,  and  represent  the  several  courses  by  CX9  C2, 
Cz,  etc.,  we  evidently  have 

lx  =  dx  cos  C\ ,  pi  =  dl  sin  Cx , 

l2  =  d2  cos  02 ,  pi  =  d2  sin  C2 , 

4  =  d3  cos  C8 ,  etc.  ps  =s  d3  sin  Cs ,  etc. 

and       [7]  =  ^  +  /2  +  4,  etc.,       [>]  =  px  +  p2  +p3,  etc. ; 

or,  as  in  the  more  simple  case  of  a  single  course, 

The  whole  difference  of  latitude  is  equal  to  the  sum  of  the 
partial  differences  of  latitude  ; 

The  whole  departure  is  equal  to  the  sum  of  the  partial  de- 
partures. 


tt 

u 

a 

(t 

a 

a 

6  NAVIGATION. 

This  applies  to  all  cases,  if  we  use  the  word  sum  in  its 
general  or  algebraic  sense. 

If  we  represent  by  Ln  the  sum  of  the  northern  diffs.  of  latitude* 
Ls  "  southern     "  " 

Pw  "  western  departures, 

JPe  "  eastern  " 

we  have  as  the  arithmetical  formulas, 

[  I  ]  =  Ln  ^^  Ls  of  the  same  name  as  the  greater, 

which  accord  with  the  usual  rules.     (Bowd.,  Arts.  115,  155.) 
The  Traverse  Form  (pp.  10  and  11)  facilitates  the  compu- 
tation. 

The  course,  (7,  and  distance  [d~\,  corresponding  to  [£]  and 
[p],  may  be  found  nearly  by  Plane  Sailing. * 

8.  The  departure  may  be  regarded  as  measured  on  the 
middle  parallel,  either  between  the  extreme  parallels  of  the 
traverse,  or  between  that  of  the  latitude  left  and  that  arrived 
at.    In  a  very  irregular  traverse  it  is  difficult  to  determine  the 

*  C  and  [d]  are  not  accurately  found,  because  [p],  the  sum  of  the 
partial  departures  of  the  traverse,  is  not  the  same  as  p,  the  departure  of 
the  loxodromic  curve  connecting  the  extremities  of  the  traverse.    Thus, 
suppose  a  ship  to  sail  from  C  to  A  by 
the  traverse  C  B,  B  A,  her  departure 
will  be  by  traverse  sailing  de  +  mn; 
whereas,  if  the  ship  sail  directly  from 
C  to  A,  the  departure  will  be  op,  which 
is  greater  or  less  than  de  +  m  n,  accord- 
ing as  it  is  nearer  to,  or  farther  from 
the  equator.      Thus  we  should  obtain 
in  the  two  cases  a  different  course  and 
distance  between  the  same  two  points. 
In  ordinary  practice,  however,  such  dif- 
ference is  immaterial. 


PARALLEL    SAILING. 


precise  parallel ;  but,  except  near  the  pole,  and  for  a  distance 
exceeding  an  ordinary  day's  run,  the  middle  latitude  suffices. 
It  is  easy,   however,  to  separate  a  traverse  into  two  or 
more  portions,  and  compute  for  each  separately. 


PARALLEL    SAILING. 

9.  The  relations  of  the  quantities  C,  d,  I,  and  p  are  ex- 
pressed in  equations  (3).  When  the  difference  of  longitude 
also  enters,  then  some  further  considerations  are  necessary, 
since  the  earth's  surface  must  now  be  regarded,  not  as  plane, 
but  spherical. 

Problem  4.     To  find  the  relations  between  — 

L,  the  latitude  of  a  parallel, 

p,  the  departure  of  two  meridians  on  that  parallel,  and 

D,  the  corresponding  difference  of  longitude. 

Solution.     In  Fig.  3,  let 
P  A  A',  PCC  be  two  meridians. 
A  C  =  p,  their  departure  on  the  paral- 
lel AC,  whose  latitude  is  AOA' 

=  O  A  B  =  Z,  and  whose  radius  is 

BA  =  n 
A'  C  =  D,  the  measure  of  A  P  C,  the 

difference  of  longitude  of  the  same 

meridians,    on   the    equator   A'  C, 

whose  radius  is  0  A'  =  0  A  =  R. 

A  C,  A'  C  are  similar  arcs  of  two  circles,  and  are  therefore 
to  each  other  as  the  radii  of  those  circles ;  that  is, 

A  C  :  A'  C  =  B  A  :  0  A',  or  p  :  D  =  r  :  B. 

In  the  right  triangle  0  B  A, 

B  A  =  0  A  x  cos  O  A  B,     or         r  =  R  cos  L  ;    (4) 


8  NAVIGATION. 

that  is,  the  radius  of  a  parallel  of  latitude  is  equal  to  the  radius 
of  the  equator  multiplied  by  the  cosine  of  the  latitude. 

Substituting  (4)  in  the  preceding  proportion,  we  obtain 

p  :  D  =  cos  Z  :  1, 

or 

p  =  D  cos  Z,  D  =p  sec  Z,  (5) 

which  express  the  relations  required.     (Bowd.,  Art.  118.) 

These  relations  may  be  graphically  represented  by  a  right 
plane  triangle  (Fig.  4),  of  which 

D  is  the  hypothenuse, 

Z,  one  of  the  angles, 

p,  the  side  adjacent  that  angle. 

The  Traverse  Table,  or  Table  of 
Bight  Triangles,  may  therefore  be  used  for  the  computation. 

MIDDLE  LATITUDE  SAILING. 

10.  Problem  5.  Given  the  course  and  distance  and 
the  latitude  left,   to  find  the  difference  of  longitude. 

Solution.     By  Plane  Sailing, 

.    I  =  d  cos  C,  p  =  d  sin  C\  (3) 

by  Arts.  2  and  6, 

Z'  =  Z  +  l,  Z0  =  t(Z'  +  Z)  =  Z  +  $l;  (6) 

and  by  equation  (5), 

D=p  sec  Z0,  (7) 

or  Z  =  d  sin  C  sec  Z0 .  (8) 

Equations  (3),  (6),  and  (7)  or  (8)  afford  the  solution  required. 

The  assumption  of  Z0  =  |  (Z'  -f-  Z),  or  the  middle  latitude, 
suffices  for  the  ordinary  distance  of  a  day's  run ;  but  for  larger 
distances,  and  where  precision  is  required,  we  should  use 
"  Mercator's  Sailing  "   (Art.  14). 


v\'i>u 


MIDDLE   LATITUDE   SAILING. 


0 


11.  Strictly,  the  middle  latitude  should  be  used  only 
when  both  latitudes,  L  and  J/9  are  of  the  same  name,  as 
is  evident  from  Fig.  1. 

If  these  latitudes  are  of  different  names,  and  the  distance 
is  small,  \  (Z  +  _Z/),  numerically,  may  be  used  ;  or  we  may 
even  take  p  =  D,  since  the  meridians  near  the  equator  are 
sensibly  parallel. 

If  the  distance  is  great,  the  two  portions  of  the  track  on 
different  sides  of  the  equator  may  be  treated  separately. 
(Art.  18.) 

When  several  courses  and  distances  are  sailed,  as  is  ordi- 
narily the  case  in  a  day's  run,  p  and  I  are  found  as  in  Trav- 
erse Sailing,  and  then  D  by  regarding  p  as  on  some  parallel 
midway  between  the  extremes  of  the  traverse.  (Art.  8.) 
(Bowd.,  Art.  155.) 


12.    The  relations   of  the   quantities   involved  in  Middle 
Latitude  Sailing,  namely, 

Oy  d,  p,  I)  X0 ,  and  D, 

are  represented  graphically  by  combining 
the  two  triangles  of  Plane  Sailing  and 
Parallel  Sailing,  as  in  Fig.  5,  in  which 


C  =  A  C  B, 
d  =  CA, 
p   =BA, 


Z0  =  BAE 

i)=AE, 


By  these  two  right  triangles,  all  the  Fig.  5. 

common  cases  classed  under  Middle  Lat- 
itude Sailing  (Bowd.,  Art.  121)  may  be  solved,  if  we  add  the 
formulas, 

Z'  =  Z  +  l  A'  =  \  +  D. 


10 


NAVIGATION. 


Example  in  Middle  Latitude  Sailing 

1.    Required  the  course  and  distance  from  San  Francisco  to 
Yokohama. 


San  Francisco,  37°  48'  N. 
Yokohama,        35°  26'  N. 


122° 28' W.  (Table  49,) 
139°  39'  E.  (    Bowd.     ( 


1=    2°  22' =  142' 
X0  =  36°  27'      1.  cos 
logD 
ar.  co.  log  I 
G  —  S.  88°  17'  W.    1.  tan  C  11.52203 
d  =  4726' 


D  =  97°  53'  =  5873' 

9.90546 

3.76886 

7.84771  log  I  2.15229 
1.  sec  C  11.52222 
log  d      3.67451 


Examples  in  Traverse  Sailing. 

13.  A  ship  from  the  position  given  at  the  head  of  each 
of  the  following  traverse  forms  sails  the  courses  and  dis- 
tances stated  in  the  first  two  columns ;  required  her  latitude 
and  longitude. 

1.    August  8,  noon  —  Lat.  by  Obs.,      35°  35'  N. 
Long,  by  Chro.,  18°  38'  W. 


Courses. 

DlST. 

N. 

s. 

E. 

W. 

/ 

/ 

/ 

/ 

/ 

N.  N.  E.  i  E. 

50 

44.1 

23.6 

S.|W. 

46.2 

45.7 

6.7 

S.  by  E.  h  E. 

16.5 

15.8 

4.8 

N.  E. 

38 

26.9 

26.9 

S.  S.  W.  i  w. 

41.8 

37.8 

17.9 

192.5 

71.0 

99.3 

55.3 

24.6 

S.  E.  i  E. 

41.5 

28.3 

30.7 

38 

=     D. 

August  9,  noon  —  Lat.     by  Acct.,  35°  7'  N. 
Long,  by  Acct.,  18°  0'  W. 


MERCATOR'S   SAILING. 


11 


2.    Apr.  23,  noon  —  Lat.  by  Obs.,         41°  31' N. 
Long,  by  Chro.,  178°  25'  W. 


Courses. 

DlST. 

N. 

S. 

E. 

W. 

S.  21°  W. 

29 

27.1 

10.4 

S.  37°  W. 

20.6 

16.5 

12.4 

S.  56°  W. 

72 

40.3 

59.7 

S.  71°  W. 

16.4 

5.3 

15.5 

S.  82°  W. 

23.7 

3.3 

23.5 

N.  88°  W. 

45 

1.6 

45 

1.6 

92.5 
90.9 

166.5 
D  =  219.8 

Apr.  24,  7  a.m.,  Lat.     by  Acct.,    40°  OO'.l  N. 
Long,  by  Acct.,  177°  55'.2  E. 

In  this  example  the  courses  are  expressed  in  degrees, 
which  is  the  preferable  method. 

MERCATOR'S     SAILING. 

14.  Middle  Latitude  Sailing  suffices  for  the  common  pur- 
poses of  navigation  ;  but  a  more  rigorous  solution  of  problems 
relating  to  the  loxodromic  curve  is  needed.  These  solutions 
come  under  "  Mercator's  Sailing." 

Problem  6.  A  ship  sails  from  the  equator  on  a  given 
course,  C,  till  she  arrives  in  a  given  latitude,  L  ;  to  find 
the  difference  of  longitude,  D. 

Solution.  Let  the  sphere  (Fig.  6)  be  projected  upon  the 
plane  of  the  equator  stereographically.  The  primitive  circle 
ABC.  .  .  .  M  is  the  equator. 

P,  its  centre,  is  the  pole  (the  eye  or  projecting  point  being 
at  the  other  pole).* 

The  radii,  P  A,  P  B,  P  C,  etc.,  are  meridians  making  the 


*  Principles  of  stereographic  projection. 


12  NAVIGATION. 

same  angle  with  each  other  in  the  projection  as  on  the  sur- 
face of  the  sphere.* 

The  distance  P  m,  of  any  point 
m  from  the  centre  of  the  projec- 
tion, =  tan  £  (90°  —  L),  the  tan- 
gent of  J  the  polar  distance  of  the 
point  on  the  surface  which  m  rep- 
resents, the  radius  of  the  sphere 
being  1.* 

This  curve  in  projection  makes 
the  same  angle  with  each  merid- 
ian as  the  loxodromic  curve  with  F1*-  6- 
each  meridian  on  the  surface.* 

A  M  is  the  whole  difference  of  Longitude  D. 

If  we  suppose  this  divided  into  an  indefinite  number  of 
equal  parts,  A  B,  B  C,  C  D,  etc.,  each  indefinitely  small,  and 
the  meridians  P  A,  P  B,  P  C,  etc.,  drawn,  the  intercepted  small 
arcs  of  the  curve  Abe.  .  .  .  m  may  be  regarded  as  straight 
lines,  making  the  angles  ~P  A  b,  ¥  b  c,  I*  c  d,  etc.,  each  equal 
to  the  course  (7;  and  consequently  the  triangles  PAi,  P  b  c, 
P  c  d,  etc.,  similar.     We  have  then 

PA:P£  =  P£:Pc=Pc:Ptf,  etc., 

or  the  geometrical  progression, 

T ,  ,,  P  A  :  P  b  :  P  c  :  .  .  .  .  P  m. 

If  then 

D  =  the  whole  difference  of  longitude, 
d  =  one  of  the  equal  parts  of  D, 

-=  will  be  the  number  of  parts,  and 
a 

-z  +  1  the  number  of  meridians  PA,  P  b. . . .  P  m, 

*  Principles  of  stereographic  projection. 


MERCATOR'S   SAILING.  13 

or  the  number  of  terms  of  the  geometrical  progression :  and, 
employing  the  usual  notation, 

the  first  term  a  =  P  A  =  1, 

the  last  term    I  =  P  m  =  tan  \  (90°  —  L), 

71  —  1  ==  —  > 

d 

the  ratio  r  =  7—7  • 

PA 

To  find  this  ratio,  we  have  in  the  indefinitely  small  right 

triangle  ABb, 

tan  B  A  b  =  cot  P  A  b  =  ^  > 

B  A 


or 

cot  C  = 

PA- 
d 

Yb 

> 

whence 

PA 

-Yb  = 

■■  d  cot  0; 

Yb  = 

:PA    - 

d  cot  C, 

and,  since 

PA  = 

1, 

r  = 

P^ 

:PA~ 

1  —  d  cot  a 

Then  by  the  formula  for  a  geometrical  progression, 

1=  ar"-1, 
(Algebra,  p.  240)  we  have 

tan  I  (90°  -  Z)  =  (l-d  cot  C)*. 
Taking  the  logarithm  of  each  member,  we  have 

log  tan  J  (90°  -  L)  =  -  log  (1  -  d  cot  C).  (9) 

But  we  have  in  the  theory  of  logarithms 

(JVaperian)  log  (1  +  n)  =  n  -r  —  -f  —  —  —  -f  etc 

and  2        8        4 

(Common)    log(l-f»)  =  ww-|  +  |-j  +  etc. . .  .1.  (10) 

in  which  the  modulus  m  =  .434294482. 


14  NAVIGATION. 

Hence,  putting        n=  —  d  cot  C, 

log  (1  -  d  cot  C)  =  m  [-  d  cot  C -  J  tf2 cot2  6r 
—  J  6?3  cot3  (?—  etc ], 

and  substituting  in  (9)  and  reducing, 

log  tan  \  (90°  -  Z)  =  -  m  X 2>  [cot  6"+  J  (7 cot2  C 
+  J  tf3cot36'  +  etc....].  (11) 

This  equation  is  the  more  accurate  the  smaller  d  is  taken, 
so  that  if  we  pass  to  the  limit  and  take  d  =  0,  it  becomes  per- 
fectly exact.  The  broken  line  Abe ....  m  then  becomes  a 
continuous  curve,  and  our  equation  (11)  becomes 

log  tan  I  (90°  -  L)  =  -  m  X  D  cot  6y; 
whence 

J=_logtanU90°-J)tanC,  } 

But  in  this  equation  D  is  expressed  in  the  same  unit  as 
tan  C,  that  is,  in  terms  of  radius.     (Pl.  Trig.,  Art.  11.) 

To  reduce  it  to  minutes  we  must  multiply  it  by  the  radius 
in  minutes,  or  /  =  3437/.74677. 

Substituting  the  value  of  m,  we  shall  have  (in  minutes), 

3437' 74677 
D  "  -  .434294482  ^  tan  *  ^90°  "  Z>  ta"  C 

To  avoid  the  negative  sign,  we  observe  that 

1  1 


tan  J  (90°  -  Z) 


cot  £  (90°  -  Z)       tan  J  (90°  +  Z)  ' 

or  that 

-  log  tan  J  (90°  -  Z)  =  log  tan  J  (90°  +  Z). 

Hence  we  have,  by  reducing, 

D  =  7915/.70447  long  tan  (45°  +  J  X)  tan  (7.  (13) 

Note.  —  Problem  6  may  be  more  readily  solved,  and  equation  (13) 
obtained  by  aid  of  the  Calculus. 


MEE  CA  TOR  '  S   SA I  LING. 


15 


In  Fig.  1,  suppose  c  a  to  be  an  ele- 
ment of  the  loxodromic  curve  C  A : 

cb  will  be  the  corresponding  element 

of  the  meridian,  and 
b  a  x  sec  L,  the  element  of  the  equator; 
L  being  the  latitude  of  the  indefinitely 

small  triangle  cab. 

By  Articles  5  and  10,  using  the  no- 
tations of  the  Calculus,  we  have 

d.L  =  cosCdd      dp  =  tanCdZ 
d  D  —  sec  L  d p  =  tan  C  sec  IdX, 


in  which  C  is  constant. 

By  integrating   the  last  equation  between  the  limits  L  =  0  and 
L  =  X,  we  shall  have 

D  =  tan  C  I     sec  L  d  i, 
Jo 

the  whole  difference  of  longitude  required  in  Problem  6. 
To  effect  the  integration,  put 

sin  L  =  x,  then  by  differentiating, 

dx 

and  multiplying  by  sec  i, 


di  = 

sec  L  d  L  = 
sec  i  d  X  = 


dx 


cosL 

dx 
cos2  £      1  —  sin2 1/ 

dx 


,  or 


1—  x2 

Resolving  into  partial  fractions,  we  obtain 

dx     ,      dx 


sec 


and 


Jo 


sec  L  d  L  =  |  [log  (1  +  x)  -  log  (1  -  x)] 


=  log 


=  .ogV/£ 


Whence  we  have 


sin  L 

=  log  tan  (45°  +  h  L)         Pl.  Trig.  (154). 
D  =  log  tan  (45°  -J-  i  L)  tan  C. 


16  NAVIGATION. 

But  in  this  the  logarithm  is  Naperian,  and  D  is  expressed  in  terms  of 

the  radius  of  the  sphere.     To  reduce  to  common  logarithms,  we  divide 

by  m  =  .434294482,  and  to  minutes  by  multiplying  by  r'  =  3437'. 74677, 

and  obtain 

T)  =  7915'. 70447  log  tan  (45°  +  $  L)  tan  C, 

as  in  (13). 

15.  Formula  (13)  is  based  upon  the  assumption  that  the 
earth  is  a  sphere.  Allowing  for  the  meridional  eccentricity 
of  the  earth,  which  according  to  Bessel  is 

\—  =  0.003342773  =  c, 

299.1528  ' 

the  formula  used  for  computing  the  table  of  meridional  parts 

1S  M  =  7915,.7045  log  tan  (45°  +  \  L) 

—  a  {f  sin  X+  J  e4  sin2  L)  (14) 


in  which      e      =  V2  c  -  c2  =  0.0816968, 

and  ae2=  22r.9448,         \ae*  =  0'.  051047. 

The  values  of  M  computed  by  (14)  for  each  minute  of  L 
from  0°  to  86°  form  the  Table  of  Meridional  Parts  or  Aug- 
mented Latitudes.     (Bowd.,  Table  3.) 

In  practice,  then,  we  have  only  to  take  the  value  of  M  cor- 
responding to  X,  and  D  is  then  found  by  the  formula, 

D  =  M  tan  C.  (15) 

M  has  the  same  name,  or  sign,  as  L. 

16.  Problem  7.  A  ship  sails  from  a  latitude,  L,  to 
another  latitude,  U ,  upon  a  given  course,  C;  find  the 
difference  of  longitude,  D. 

Solution.    Let 

M  be  the  augmental  latitude  corresponding  to  X, 
M'     "  "  "  "  Lf. 


MERCATOR'S   SAILING. 


17 


The  difference  of  longitude  from  the  point,  A,  where  the 

track  crosses  the  equator  to  the  first  position,  whose  latitude 

is  L.  will  be 

DJ  =  M  tan  C; 

and  to  the  second  position,  whose  latitude  is  X', 

Dtt  —  M'  tan  0\ 

and  we  shall  have 

D  =  Du  -  D4  =  (M'  -  M)  tan  C\  (16) 

or,  when  M '  <  M, 

B  =  2>rf  -  2>y/  =  (Jf-  IT)  tan  C; 

since  the  sign  of  D  is  determined  by  the  course. 

If  L  and  Z'  are  of  different  names,  so  also  are  M  and  M' , 
and  we  have  numerically 

&=(M+M')  tan  £ 

17.  The  difference,  J!/'  —  J/,  is  called  the  meridional,  or 
augmented,  difference  of  latitude.  Representing  this  by  m,  we 
have 


D  =  m  tan  C. 

The  relation  between  these  quantities 
is  represented  by  a  plane  right  triangle 
(Fig.  7),  in  which 

C  is  one  of  the  angles, 

m  =  C  E,  the  side  adjacent, 

i)=EF,  the  side  opposite. 

The  triangle  of  "  Plane  Sailing "  has 
the  same  angle  C,  with 

I  =  C  B,  the  adjacent  side, 
and       p  =  B  A,  the  opposite  side. 


(17) 


Pig.  7. 


18  NAVIGATION. 

Fig.  7  represents  these  two  triangles  combined.  By  them, 
all  the  common  cases  under  Mercator's  Sailing  can  be  solved, 
either  by  computation  or  by  the  Traverse  Table.  (Bowd., 
Art.  128.) 

The  relations  between  the  several  parts  involved  are 

I    =  d  cos  C,  L'  =  L+  I, 

p  —  d  sin  C,  m  =  M'  —  M, 

D  =  m  tan  C,  a'  =  A  +  D ;  (18) 

and  since  p  =  I  tan  C, 

I :  m  =  p  :  D. 

18.  Problem  8.  Given  the  latitudes  and  longitudes 
of  two  places,  find  the  course,  distance,  and  departure. 
(Bowd.,  Art.  128,  Case  I.) 

Solution.  L  and  U  being  given,  we  take  from  Table  3  M 
and  M '. 

We  have  I  =  U  -  Z,         m  =  M'  -  M,         D  =  \'  -  X  ; 
by  Mercator's  Sailing,  tan  C  =  —  ; 

and  by  Plane  Sailing,         d=  I  sec  C,        p  =  1  tan  C; 

?,  m,  and  C  are  north  or  south  according  as  U  is  north  or 
south  of  JL. 

D,  jo,  and  C  are  east  or  west,  according  as  \r  is  east  or  west 
of  A. 

If  the  two  places  are  on  opposite  sides  of  the  equator,  we 
have  numerically 

l=Z'+Z,  m  =  M'  +  M. 

Examples. 

1.  Kequired  the  course  and  distance  from  Cape  Frio  to 
Lizard  Point,  England. 


MERCATOR'S   SAILING.  19 

Cape  Frio, 

L  =  23°  Or  S.      A.  =  42°  00'  W.     M  =  1410.7  S. 
Lizard  Point, 

2/ =  49°  58'  N.      X'=    5°12'W.     Jf' =  3453.8  N.    log-  2)  =  3.34400 

J  =  72°  59'  N.      D  =  36°48/  E.      m    =  4864.5  N.    log  m  =  3.68704 

C  =  N.  24°  24'  48"  E.      1.  sec  O  =  0.04068  1.  tan  C  =  9.65696 

logZ    =  3.64137 
d   =  4809'.  log  d  =  3.6S205 

2.    Required  the  course  aud  distance  from  San  Francisco  to 
Yokohama. 

San  Francisco, 

37°  48'  N.  122°  28'  W.      M  =  2439 
Yokohama, 

35° 26'  N.  139° 39'  E.       M'  =  2262.8        log  D  =    3.76886 
I  =  2°  22'  S.  B  =  97°  53'  E.  =  5873'  m  =    176.2  co.  log  m  =    7.75399 

C  =  N.  91°  43'  W.  1.  sec  =  11.52308            1.  tan  C  =  11.52285 

or,            S.  88°17/W.  log  I  =    2.15229 

d  =  4735.5  log  d  =    3.67537 

19.  The  loxodromic  curve  on  the  surface  of  the  earth  and 
its  stereographic  projection  (Fig.  6)  present  a  peculiarity 
worthy  of  notice.  Excepting  a  meridian  and  parallel  of  lati- 
tude, a  line  which  makes  the  same  angle  with  all  the  merid- 
ians which  it  crosses  would  continually  approach  the  pole, 
until,  after  an  indefinite  number  of  revolutions,  the  distance 
of  the  spiral  from  the  pole  would  become  less  than  any  as- 
signable quantity.  It  is  usual  to  say  that  such  a  curve  meets 
the  pole  after  an  infinite  number  of  revolutions.  Still,  how- 
ever, it  is  limited  in  length. 

For  we  have  for  the  length  of  any  portion, 

by  Plane  Sailing,         d  =  (L'  —  X)  sec  C. 

If  L  =  0  and  If  =  90°  =  -, 

2' 

the  whole  spiral  from  the  equator  to  the  pole  will  be,  with 
radius  =  1, 


20  NAVIGATION. 

d  =  \  sec  <7. 

If  i  =  -  90°  =  -  *  and  L'  =  90°  =  £, 

we  have,  as  the  entire  length  from  pole  to  pole, 

d—TT  sec  C. 

If  also  (7=0,  or  the  loxodromic  curve  is  a  meridian,  d  —  *,  a 
semicircumference,  as  it  should  be. 

So  also  the  length  of  the  projected  spiral  Abe...  (Fig. 
6)  from  A  to  m  can  readily  be  shown  to  be  (calling  this 
length  8) ; 

8  =  Jf m  sec  (7=  [1  —  tan  (45°  —  j-X)]  sec  (7, 

/-r.     m  -.,--.  x     *      2  tan  4  L  sec  (7 

or  (Pi.  Tbio.,  151),    8-     1+*tanii-     ; 

and  its  length  from  the  equator  to  the  pole,  taking  X  =  90°, 

8  =  sec  C. 


A    MERCATOR'S    CHART. 

20.  On  a  Mercator's  chart,  the  equator  and  parallels  of 
latitude  are  represented  by  parallel  straight  lines ;  and  the 
meridians  also  by  parallel  straight  lines  at  right  angles  with 
the  equator.  Two  parallels  of  latitude,  usually  those  which 
bound  the  chart,  are  divided  into  equal  parts,  commencing  at 
some  meridian  and  using  some  convenient  scale  to  represent 
degrees,  and  subdivided  to  10',  2',  V,  or  some  other  convenient 
part  of  a  degree,  according  to  the  scale  employed. 

Two  meridians,  usually  the  extremes,  are  also  divided  into 
degrees,  and  subdivided  like  the  parallels  of  latitude,  but  by 
a  scale  increasing  constantly  with  the  latitude  :  so  that  any 
degree  of  latitude  on  such  meridian,  instead  of  being  equal  to 
a  degree  of  the  equator,  is  the  augmented  degree,  or  augmented 


A  MERCATOR'S   CHART.  21 

difference  of  1°  of  latitude,  derived  from  a  table  of  "  meridi- 
onal parts."  (Bowd.,  Table  3.)  The  meridian  is  graduated 
most  conveniently  by  laying  off  from  the  equator  the  aug- 
mented latitudes  ;  or  from  some  parallel,  the  augmented  differ- 
ence of  latitude  for  each  degree  and  part  of  a  degree,  —  using 
the  same  scale  of  equal  parts  as  for  the  equator. 

21.  As  on  other  maps  and  charts,  parallels  of  latitude  and 
meridians  are  drawn  at  convenient  intervals  ;  places,  shore 
lines  of  continents  and  islands,  harbors  and  rivers,  etc.,  are 
plotted,  each  point  in  its  proper  position ;  and  such  configu- 
rations of  the  land  represented  as  the  purpose  of  the  map 
requires. 

22.  To  plot  on  a  chart  a  point  whose  latitude  and  longitude 
are  given.  By  means  of  the  scales  at  the  sides,  draw  a  paral- 
lel of  latitude  in  the  latitude,  and  by  means  of  the  scales  at 
the  top  or  bottom,  a  meridian  in  the  longitude  of  the  point ; 
or  so  much  of  each  as  suffices  to  find  their  intersection. 

23.  In  nautical  charts  the  soundings  in  shoal  water  are 
put  down,  and  even  the  character  of  the  bottom ;  and  on  those 
of  a  large  scale,  also,  the  contour  lines  of  the  bottom,  or  lines 
of  equal  depth.  The  variation  of  the  compass  at  convenient 
intervals,  and  lines  of  equal  variation,  are  valuable  additions. 

24.  The  meridians  on  this  chart  being  parallel,  arcs  of  par- 
allels of  latitude  are  represented  as  equal  to  the  corresponding 
arcs  of  the  equator :  thus  each  is  expanded  in  the  proportion 
of  the  secant  of  its  latitude  to  1 ;  as  is   evident  from  the 

formula 

J)  =  p  sec  L. 

It  can  be  shown  that  very  small  portions  of  the  meridians 


22 


NAVIGATION. 


are  expanded  in  the  same  proportion ;  as  for  example,  a  de- 
gree whose  middle  latitude  is  60°  is  120',  or, 

60'  of  the  equator  x  sec  60°. 
But  the  two  half  degrees  are  unequally  expanded ;  for 

from  59£°  to  60°  is  represented  by  59', 
"    60°    to  60£°         "  "       61',  nearly. 

A  small  circle  on  the  surface  of  the  earth  of  1°  diameter  at 
the  equator  is  then  represented  by  a  circle,  whose  diameter  is 

1°; 

in  lat.  30°  nearly  by  a  circle,  whose  diameter  is  1°  X  sec  30°, 
"      60°       "  "  "  "  1°  X  sec  60°, 

"      Z         "  "  «  "  1°  X  sec  X; 


but  not  exactly  by  a  circle,  since  the  meridians  are  augmented 
more  rapidly  as  the  latitude  is  greater. 

Such  a  chart,  then,  while  representing  a  narrow  belt  at  the 
equator  in  proper  proportions,  presents  a  view  of  the  earth's 
surface  expanded  at  each  point,  both  in  latitude  and  longi- 
tude, proportionally  to  the  secant  of  its  latitude. 


25.  If  we  take  any  two  points,  C  F,  on 
this  chart,  and  join  them  by  a  straight 
line,  and  form  a  right  triangle  by  a  merid- 
ian through  one,  and  a  parallel  of  latitude 
through  the  other,  we  shall  have  the  tri- 
angle of  Mercator's  sailing  (Fig.  8)  :  for  the 
intercepted  portion  of  the  meridian,  C  E, 
is  the  augmented  difference  of  latitude ; 
and  of  the  parallel  of  latitude,  E  F,  is  the 
difference  of  longitude.  Hence  the  angle 
E  C  F  is  the  course.    (Art.  17.)    Moreover, 


Fig.  8. 


A  MERCATOR'S   CHART.  23 

the  loxodromic  curve  is  represented  by  the  straight  line  C  F ; 
for  if  we  take  any  intermediate  point  of  this  curve,  and  let  d 
be  its  position  on  the  chart,  d  must  be  in  the  line  C  F,  other- 
wise when  we  construct  the  triangle  of  Mercator's  Sailing  we 
shall  have  an  angle  at  C  different  from  E  C  F,  the  course, 
which  for  every  point  of  the  loxodromic  curve  is  the  same. 

Thus  a  Mercator's  chart  presents  two  decided  advantages 
for  nautical  purposes ;  viz., 

1.  The  ship's  track  is  represented  by  a  right  line. 

2.  The  angle  which  this  line  makes  with  each  meridian 
is  the  course. 

To  find  the  course  from  one  point  to  another  on  the  chart, 
all  that  is  necessary  is  to  draw  a  line,  or  lay  down  the  edge 
of  a  ruler,  through  the  two  points,  and  measure  its  angle 
with  any  meridian.  A  convenient  mode  is  to  refer  such  line 
by  means  of  parallel  rulers  to  the  centre  of  one  of  the  com- 
pass diagrams,  which  usually  will  be  found  on  the  chart,  and 
reading  the  course  from  the  diagram. 

When  such  diagrams  are  constructed  with  reference  to  the 
true  meridian,  the  course  obtained  is  the  true  course,  and  not 
the  magnetic  course. 

26.  The  distance  C  F,  however,  is  an  augmented  distance, 
which  we  may  measure  nearly  by  the  augmented  scale  on  the 
meridians  of  the  chart  (the  middle  latitude  of  the  scale  used 
being  the  same  as  that  of  the  line  C  E).  Or  we  may  con- 
struct the  proper  distance,  CA,  by  constructing  the  triangle, 
C  B  A,  of  Plane  Sailing,  in  which  C  B  is  the  proper  difference 
of  latitude,  the  scale  for  which  is  on  the  equator. 

The  distance  here  spoken  of,  though  represented  on  this 
chart  by  a  straight  line,  is  not  the  shortest  distance  between 
the  two  points ;  for  on  the  surface  of  a  sphere,  the  shortest 


24  NAVIGATION. 

distance  between  two  points  is  the  arc  of  a  great  circle  which 
joins  them.     To  find  this  belongs  to  great-circle  sailing. 

27.  In  Polyconic  Projection  each  parallel  of  latitude  is 
developed  upon  its  own  cone,  the  vertex  of  which  is  on  the 
axis  at  its  intersection  with  the  tangent  to  the  meridian  at 
the  parallel.  The  advantage  of  a  chart  so  constructed  is  that 
those  portions  lying  near  the  central  meridian  will  be  but 
little  distorted. 

The  method  of  construction,  together  with  'tables  for  the 
Polyconic  as  well  as  Mercator's  Projection,  are  given  in  Pro- 
jection Tables  for  the  use  of  the  United  States  Navy.  (Bur. 
Navgn.). 

great-circle  sailing. 

28.  The  rhumb-line,  or  spiral  curve,  which  cuts  all  the 
meridians  at  the  same  angle,  was  used  formerly  by  navi- 
gators in  passing  from  point  to  point  on  account  of  the 
simplicity  of  the  calculations  required  in  practice.  But,  as 
has  been  stated,  it  is  a  longer  line  than  the  great  circle 
between  the  same  points,  and  therefore  the  intelligent  navi- 
gators of  the  present  day  are  substituting  the  latter  wherever 
practicable. 

On  the  Mercator  chart,  however,  the  arc  of  a  great  circle 
joining  two  points,  not  on  the  equator  or  on  the  same  me- 
ridian, will  not  be  projected  into  a  straight  line,  but  into  a 
curve  longer  than  the  Mercator  distance,  and  still  greater 
than  the  distance  on  a  rhumb-line.  Hence  it  is  an  objection 
to  the  Mercator  chart,  that  the  shortest  route  from  point  to 
point  appears  on  it  as  a  circuitous  one ;  and  this  is,  doubtless, 
one  main  reason  why  merely  practical  men  have  made  so 
little  use  of  the  great  circle.     Many  of  those  unacquainted 


GREAT-CIRCLE  SAILING. 


25 


with  the  mathematical  principles  of  the  subject  are  unable 
to  comprehend  how  the  apparently  circuitous  path  on  then- 
chart  should  actually  be  the  line  of  shortest  distance. 

29.  Problem  9.  To  project  on  a  chart  the  arc  of  a 
great  circle  joining  two  given  points  on  the  globe. 

Solution.  It  will  be  necessary  to  project  a  number  of 
points  of  the  arc,  and  trace  through  these  points  the  curve 
by  hand.  To  project  a  point  on  the  chart,  we  must  know  its 
latitude  and  longitude. 

The  two  given  points,  A  and  B 
(Fig.  9),  and  the  pole,  P,  are  the 
three  angular  points  of  a  spher- 
ical triangle,  formed  by  the  arcs 
joining  these  points  with  each 
other  and  with  the  pole.  If  from 
P  we  draw  P  C0  perpendicular  to 
A  B,  the  point  C0  is  nearer  the 
pole  than  any  other  point  of  A  B ; 
that  is,  it  is  the  point  of  maximum 

latitude.  This  point  of  greatest  latitude  is  called  the  vertex 
of  the  great  circle. 

To  find  the  latitude  and  longitude  of  this  vertex. 

This  may  be  done  by  a  direct  application  of  the  rules  of 
Spherical  Trigonometry,  first  finding  the  angles  A  and  B  by 
Case  I.  of  Sph.  Trig.,  and  then  solving  one  of  the  right  tri- 
angles A  P  O0  or  B  P  C0.  But  in  practice  the  following 
method  is  preferable. 

Let  Zx  =  (90°  —  P  A),  and  \x  be  the  latitude  and  longitude 
of  A,  the  point  left. 
X2  =  (90°  —  P  B),  and  X2,  be  the  same  of  B,  the  point 
arrived  at. 


Fig.  9. 


26  NAVIGATION. 

Lv  =  (90°  —  P  C0),  and  \v  be  the  same  of  the  vertex,  C0. 
A    =  (Ax  —  A2),  is  the  difference  of  longitudes  of  A  and  B. 

A  B  =  d,  is  the  distance  between  A  and  B.  Draw  K  per- 
pendicular to  P  A,  dividing  it  into  Y  D  =  <f>  and  A  D  =  90° 

-  (A  +  +). 

Then  in  the  triangles  A  B  D,  and  B  D  P,  by  Napier's  Rules 
(Sph.  Trig.,  Art.  46)  we  have 

cos  A.  =  tan  <f>  tan  Z2  or,  tan  <f>  =  cos  A  cot  Z2  (19) 

sin  <f>  as  cot  A  tan  K  (20) 
cos  (Lx  +  <f>)  =  cot  A  tan  K  or, 

cot  A  =  cos  (Xx  -f-  <f>)  cot  A  cosec  <f>.  (21) 

A  is  the  course  from  A. 

In  the  right  triangle  P  C0  A,  we  have 

cos  Lv  —  cos  Lx  sin  A  (22) 
sin  Lx  =  cot  A  cot  (Ax  —  Xv)  or, 

cot  (Ax  —  Xc)  =  sin  Xx  tan  ^1.  (23) 

sin  d  sin  ^4  =  cos  Z2  sin  A  (check).  (24) 

30.  To  find  any  number  of  points,  C,  C",  C",  etc.,  C1?  C2, 
C8,  etc.,  we  may  assume  at  pleasure  the  differences  of  longi- 
tude from  the  vertex  C0  P  C ,  C0  P  C",  C0  P  C",  etc.  It  is  best 
to  assume  them  at  equal  intervals  of  5°  or  10°. 

Let     A'    =  C0  P  C,  U    =  (90°  -  P  C),  the  lat.  of  C, 

X"  =  C0  P  C",  X"  =  (90°  -  P  C"),  «         C", 

X'"  =  C0  P  C",  2/"  =  (90°  -  P  C"),  "         C" 
etc.                                etc. 

then  the  right  triangles  C0  P  C,  C0  P  C",  C0  P  C",  etc.,  give 

tan  U    =  tan  Z„  cos  A',  \ 

tan  Z"  ==  tan  Lv  cos  A",  I  (25) 

tan  If"  =  tan  Zr  cos  A'",  etc.] 


GREAT-CIRCLE  SAILING.  27 

Or  we  may  assume  values  of  L',  L" ',  Lf",  etc.,  and  find  the 
corresponding  values  of  a',  a",  a'",  etc.,  by  the  formulas 

cos  A/    =  tan  U    cot  Zv,        "I 

cos  A"  =  tan  L"  cot  Lv,         \  (26) 

cos  Xffr  =  tan  U"  cot  Lv,  etc.  J 

from  which  we  shall   have  two  values  of   X  for  each  value 
of  L. 

Having  thus  found  as  many  points  as  may  be  deemed 
sufficient,  we  may  plot  them  upon  the  chart,  and  through 
them  trace  the  required  curve. 

31.  Problem  10.  To  find  the  great-circle  distance  and 
course  between  two  given  points. 

Solution.  Let  d  be  the  distance  between  the  two  points  A 
and  B  (Fig.  9). 

Then  in  the  triangles  BDP  and  ADB,  by  Napier's 
Rules,  we  have, 

sin  X2  =  cos  <£  cos  K,  (27) 

cos  d    =  sin  (Lx  -\-  <f>)  cos  K,  (28) 

cos  d    =  sin  (B1  +  <£)  sin  X2  sec  <£,  (29) 

cot  d    =  cos  A  tan  (Lx  -\-  <f>).     (Check.)  (30) 

d,  reduced  to  minutes,  will  be  the  distance  in  geographic  miles. 

The  course  from  A  is  found  by  (21). 

The  course   from  B  may  be  found  from  the  right   triangle 

B  &***>  cos  B  =  sin  Zv  sin  (X,  —  A2).  (31) 

The  vertex  lies  between  A  and  B,  unless  either  A  or  B  is 
>90°. 

32.  Example.  To  find  the  great  circle  from  San  Fran- 
cisco to  Yokohama.     (Formulas  19,  21,  22,  23,  25,  29.) 


28 


NAVIGATION. 


San  Francisco, 
Lat.  Lx  — 
Yokohama, 


37  48  N.  Long.  122  28  W. 

35  26  N.  139  39  E. 

97  53      cos  — 9.13722      cot  —  9.14134 


L2  =    35  26       i 

cot      0.14870 

sin 

9.76324 

Lx  =    37  48 

<f>'  =  169°  05' 21 

"  tan  9.28502 

cosec  0.72290    sec  - 

0.00792 

£i 

+  <f>  =206°  53'  21 

cos  — 9.95031    sin- 

9.65540 

C  =  N.  56°52/40/,W.                  cot      9.81455 

d    =        74o30'44"  =  4470'.75                              cos 

9.42656 

Lj  =    37  48 

cos  9.89771 

sin  9.78739 

C   =    56  52  40 

sin  9.92299 

tan  0.18545 

Lv  =    48  33  55 
L  -  Xv  =    46  47  26 

cos  9.82070 

\ 

cot  9.97284 

K  =  169  15  26  W. 

Long 

from 

Vebte 

l.COS  /.     1.  TAN  L. 
X. 

Latitude. 

Longitudes. 

0 

0 

0.00000    0.05421 

48  34    "  N. 

169  15  W.     169  15  W. 

(Vertex.) 

±    5 

9.99834    0.05255 

48  27  30 

164  15            174  15 

±10 

9.99335     0.04756 

48  08 

159  15           179  15  W. 

±15 

9.98494    0.03915 

47  35 

154  15           175  45  E. 

±20 

9.97299    0.02720 

46  48 

149  15           170  45 

±25 

9.95728    0.01149 

45  45  30 

144  15           165  45 

±30 

9.93753     9.99174 

44  27  30 

139  15           160  45 

±  35 

9.91336    9.96757 

42  52 

134  15           155  45 

±40 

9.88425     9.93846 

40  57 

129  15           150  45 

±45 

9.84949     9.90370 

38  42 

124  15           145  45 

±50 

9.80807     9.86228 

36  04 

119  15  W.     140  45  E. 

Course  N.  56° 

52' 40"  W. 

from  San  Francisco. 

Distance  =  44 

70|  miles. 

Distance  by  Mercator's  Sailing  =  4735J  miles. 

33.  To  follow  a  great  circle  rigorously  requires  a  contin- 
ual change  of  the  course.  As  this  is  difficult,  and  indeed  in 
many  cases  is  practically  impossible,  on  account  of  currents, 


GREAT-CIRCLE  SAILING.  29 

adverse  winds,  etc.,  it  is  usual  to  sail  from  point  to  point  by- 
compass,  thus  making  rhumb-lines  between  these  points. 

When  the  ship  has  deviated  from  the  great  circle  which 
it  was  intended  to  pursue,  it  is  necessary  to  make  out  a  new 
one  from  the  point  reached  to  the  place  of  destination.  It  is 
a  waste  of  time  to  attempt  to  get  back  to  an  old  line. 

34.  As  the  course,  in  order  to  follow  a  great  circle,  is 
practically  the  most  important  element  to  be  determined, 
mechanical  means  of  doing  it  have  been  devised.  Towson's 
Tables  and  Bergen's  Tables  are  used  by  English  navigators. 

Charts  are  constructed  by  a  gnomonic  projection,  on  which 
great  circles  are  represented  by  straight  lines ;  but  by  these, 
computation  is  necessary  to  find  the  course. 

35.  A  great  circle  between  two  points  near  the  equator, 
or  near  the  same  meridian,  differs  little  from  a  loxodromic 
curve.  But  when  the  differences  both  of  latitude  and  of 
longitude  are'  large,  the  divergence  is  very  sensible.  It  is 
then  that  the  great  circle,  as  the  line  of  shortest  distance, 
is  preferred. 

But  it  is  to  be  noted  that  in  either  hemisphere  the  great- 
circle  route  lies  nearer  the  pole,  and  passes  into  a  higher 
latitude,  than  the  loxodromic  curve.  Should  it  reach  too 
high  a  latitude,  it  is  usually  recommended  to  follow  it  to 
the  highest  latitude  to  which  it  is  prudent  to  go,  then  follow 
that  parallel  until  it  intersects  the  great  circle  again. 

36.  A  knowledge  of  great-circle  sailing  will  often  enable 
the  navigator  to  shape  his  course  to  better  advantage.  Let 
A  B  (Fig.  10)  be  the  loxodromic  curve  on  a  Mercator's  chart, 
A  C  B  the  projected  arc  of  a  great  circle. 

The  length  on  the  globe  of  the  great  circle  A  C  B  is  less 


30 


NAVIGATION. 


than  that  of  the  rhumb-liue  A  B,  or  of  any  other  line,  as 
ADB,  between  the  two.     But  A  C  B  is  also  less  than  lines 

that  may  be  drawn  from  A  to  B  on 
the  other  side  of  it,  that  is,  nearer 
the  pole ;  and  there  will  be  some 
line,  as  A  D'  B,  nearer  the  pole  than 
the  great  circle,  and  equal  in  length 
to  the  rhumb-line.  Between  this  and 
the  rhumb-line  may  be  drawn  curves 
from  A  to  B,  all  less  than  the  rhumb- 
line.  If  the  wind  should  prevent 
the  ship  from  sailing  on  the  great 
circle,  a  course  as  near  it  as  practicable  should  be  selected. 
If  she  cannot  sail  between  A  B  and  A  C,  there  is  the  choice 
of  sailing  nearer  the  equator  than  A  B,  or  nearer  the  pole 
than  A  C.  The  ship  may  be  nearing  the  place  B  better  by 
the  second  than  by  the  first,  although  on  the  chart  it  would 
appear  to  be  very  far  off  from  the  direct  course. 


Fig.  10. 


37.  This  may  be  strikingly  illustrated  by  the  extreme 
case  of  a  ship  from  a  point  in  a  high  latitude  to  another  on 
the  same  parallel  180°  distant  in  longitude.  The  great-circle 
route  is  across  the  pole,  while  the  rhumb-line  is  along  the 
small  circle,  the  parallel  of  latitude,  east  or  west;  the  two 
courses  differing  90°.  Any  arc  of  a  small  circle  drawn  be- 
tween the  two  points,  and  lying  between  the  pole  and  the 
parallel  of  latitude,  will  be  less  than  the  arc  of  the  parallel. 
Hence  the  ship  may  sail  on  one  of  these  small  circles  nearly 
west,  and  make  a  less  distance  than  on  the  Mercator  rhumb, 
or  parallel  due  east.  This  is,  indeed,  an  impossible  case  in 
practice,  but  it  gives  an  idea  of  the  advantage  to  be  gained 
in  any  case  by  a  knowledge  of  the  great-circle  route. 


GREAT-CIRCLE  SAILING.  31 

It  is  possible  in  high  latitudes  that  a  ship  may  have  such 
a  wind  as  to  sail  close-hauled  on  one  tack  on  the  rhumb-line, 
and  yet  be  approaching  her  port  better  by  sailing  on  the 
other  tacky  or  twelve  points  from  the  rhumb-line  course. 

38.  The  routes  between  a  number  of  prominent  ports  rec- 
ommended by  Captain  Maury  are  mainly  great-circle  routes, 
modified  in  some  cases  by  his  conclusions  respecting  the 
prevailing  winds. 


32  NAVIGATION. 


CHAPTER   II. 

REFRACTION.  — DIP    OF    THE    HORIZON.— 
PARALLAX.  — SEMIDIAMETERS. 

REFRACTION. 

39.  It  is  a  fundamental  law  of  optics,  that  a  ray  of  light 
passing  from  one  medium  into  another  of  different  density 
is  refracted,  or  bent  from  a  rectilinear  course.  If  it  passes 
from  a  lighter  to  a  denser  medium,  it  is  bent  toward  the  per- 
pendicular to  the  surface  which  separates  the  two  media ;  if 
it  passes  from  a  denser  to  a  lighter  medium,  it  is  bent  from 
that  perpendicular.     Let 

p   •  M  and  N  (Fig.  11)  represent  two  me- 

dia each  of  uniform  density,  but 
the  density,  or  refracting  power, 
of  N  being  the  greater,- 
a  b  c,  the  path  of  the  ray  of  light 

through  them ; 
P  #,  the  normal  line,  or  perpendicu- 
lar, to  the  separating  surface  at  b. 

If  a  b  is  the  incident  ray,  b  c  is  the  refracted  ray ;  P  ba 
is  the  angle  of  incidence  ;  P  b  ar  is  the  angle  of  refraction. 

If  c  b  is  the  incident  ray,  b  a  is  the  refracted  ray,  and 
P  b  a!  and  Y  ba  are  respectively  the  angles  of  incidence  and 
refraction. 

Moreover,  these  angles  are  in  the  same  plane,  which,  as  it 


M 


N 


c 
Fig.  11. 


BEFRACTION. 


33 


passes  through  P  b,  is  perpendicular  to  the  surface  at  which 
the  refraction  takes  place ;  and  we  have  for  the  refraction 


or  the  difference  of  direction  of  the  incident  and  refracted  rays. 
A  more  complete  statement  of  the  law  for  the  same  two 
media  is,  that 

— — — — — -  =  m,  a  constant  for  these  media : 
sin  P  b  a' 

or,  the  sines  of  the  angles  of  iyicideyice  ayid  refraction  are  in  a 
constant  ratio. 

This  law  is  also  true  when  the  surface  is  curved  as  well  as 
when  it  is  a  plane. 

40.  If  the  medium  N,  instead  of  being  of  uniform  density, 
is  composed  of  parallel  strata,  each  uniform  but  varying  from 
each  other,  the  refracted  ray  b  c  will 
be  a  broken  line;  and  if,  as  in  Fig.  12, 
the  thickness  of  these  strata  is  inde- 
finitely small,  and  the  density  gradu- 
ally increases  in  proceeding  from  the 
surface  b,  b  c  will  become  a  curved 
line.  But  we  shall  still  have  for  any 
point  c  of   this   curve,  c  a!  being  a 

tangent  to  it,  .    ._,  7 

sin  P  b  a 

m, 


Fig   12. 


sin  P'  c  a! 

a  constant  for  the  particular  stratum  in  which  c  is  situated. 

This  law,  which  is  true  for  strata  in  parallel  planes,  ex- 
tends also  to  parallel  spherical  strata,  except  that  the  normals 
P  b,  P'  c  are  no  longer  parallel,  but  will  meet  at  the  centre  of 
the  sphere.  But  the  refraction  takes  place  in  the  common 
plane  of  these  two  normals. 


34 


NAVIGATION. 


41.  The  earth's  atmosphere  presents  such  a  series  of  par- 
allel spherical  strata,  denser  at  the  surface  of  the  earth,  and 
decreasing  in  density,  until  at  the  height  of  fifty  miles  the 
refracting  power  is  inappreciable.   . 

In  Fig.  13,  the  concentric  circles  M  N  represent  sections 
of  these  parallel  strata,  formed  by  the  vertical  plane  passing 
through  the  star  S  and  the 
zenith  of  an  observer  at  A. 
The  normals  C  A  Z  at  A, 
and  C  B  E  at  B,  are  in  this 
vertical  plane.  S  B,  a  ray 
of  light  from  the  star  S, 
passes  through  the  atmo- 
sphere in  the  curve  B  A,  and 
is  received  by  the  observer 
at  A. 

Let  AS'  be  a  tangent  to 
this  curve  at  A ;  then  the 
apparent    direction    of    the 

star  is  that  of  the  line  A  S'  j  and  the  astronomical  refraction 
is  the  difference  of  directions  of  the  two  lines  B  S  and  A  S'. 
This  difference  of  directions  is  the  difference  of  the  angles 
E  B  S,  E  D  S',  which  the  lines  S  B,  S'A,  make  with  any  right 
line  C  B  E,  which  intersects  them.  If,  then,  r  represent  the 
refraction,  we  have 

r=EBS-EDS'. 


Fig.  13, 


Also,  E  B  S  is  the  angle  of  incidence,  and  Z  A  S',  the  appar- 
ent zenith  distance,  is  the  angle  of  refraction  j  and  we  have 


sin  E  B  S 
sin  Z  AS' 


m 


a  constant  ratio  for  a  given  condition  of  the  atmosphere  and 


REFRACTION,  35 

a  given  position  of  A ;  but  varying  with  the  density  of  the 
atmosphere,  and  for  different  elevations  of  A  above  the  sur- 
face. For  a  mean  state  of  the  atmosphere  and  at  the  surface 
of  the  earth,  experiments  give  m  =  1.000294. 

The  principles  of  Arts.  39  and  40,  applied  to  this  case, 
show  that  astronomical  refraction  takes  place  in  vertical 
planes,  so  as  to  increase  the  altitude  of  each  star  without 
affecting  its  azimuth.  The  refraction  must  therefore  be  sub- 
tracted from  an  observed  altitude  to  reduce  it  to  a  true  alti- 
tude;  or 

h  =  h  —  r, 

in  which  h    is  the  true  altitude, 

h',  the  apparent  altitude, 
r,    the  refraction. 

These  laws  are  here  assumed.  The  facts  and  reasoning  on 
which  they  depend  belong  to  works  on  Optics.  (Bowd.,  Art. 
248.) 

42.  After  a  profound  investigation  of  the  problem,  Laplace 
obtained  a  complicated  formula  for  determining  the  refrac- 
tion. Bessel  has  modified  and  improved  Laplace's  formula. 
His  tables  of  refraction  are  now  considered  the  most  reliable. 
They  are  found  in  a  convenient  form  for  nautical  problems  in 
Table  20,  Bowditch.  The  mean  refractions  in  this  table  are 
for  the  height  of  the  barometer  30  inches,  and  the  temperature 
50°  Fahrenheit.2* 

43.  Tables  21  and  22,  Bowditch,  contain  corrections  to  be 
applied  to  the  normal  refraction  for  changes  in  temperature 
and  barometric  height,  deduced  also  from  Bessel's  Tables. 

*  Chauvenet's  Astronomy,  I,  127-172,  contains  a  thorough  investi- 
gation of  the  problem  of  refraction,  especially  of  Bessel's  formulas. 


36 


NAVIGATION. 


44.  When  h  =  90°,  or  the  object  is  in  the  zenith,  r  —  0 ; 
that  is,  the  path  is  a  straight  line. 

When  h  =  0,  or  the  object  is  in  the  horizon,  the  ray  of 
light,  nearly  horizontal,  describes  near  the  earth's  surface  a 
curve  which  is  approximately  the  arc  of  a  circle  whose  radius 
is  seven  times  the  radius  of  the  earth ;  or, 

E'  =  7R. 
This,  however,  is  in  a  mean  condition  of  the  atmosphere. 
The  curve  is  greatly  varied  in  extraordinary  states  of  the  at- 
mosphere, or  by  passing  near  the  earth's  surface  of  different 
temperatures ;  in  very  rare  cases  even  to  the  extent  of  becom- 
ing convex  to  the  surface  a  short  distance. 


DIP    OF    THE    HORIZON. 

45.    Problem  11.     To  find  the  dip  of  the  horizon. 

Solution.    Let  A  (Fig.  14)  be  the  position  of  the  observer 
at  the  height  BA  =  A,  above  the 
level  of  the  sea ;  A  H,  perpendicu- 
lar to  the  vertical  line,  C  A,  repre- 
sents the  true  horizon. 

The  most  distant  point  of  the 
horizon  visible  from  A  is  that  at 
which  the  visual  ray,  H"  A,  is  tan- 
gent to  the  earth's  surface. 

The  apparent  direction  of  H" 
is  AH',  the  tangent  to  the  curve 
A  H"  at  A.  AH  =  HAH'  is  the 
dip  of  the  horizon  to  be  found. 

Let  C  be  the  centre  of  the  earth, 

C,  the  centre  of  the  arc  H"  A. 
H",  C,  C,  are  in  the  same  straight  line,  since  the  arcs  H"  B, 


Fig.  14. 


DIP   OF   THE  HORIZON.  37 

H"  A  are  tangent  to  each  other  at  H", 

C  A,  C'  A,  are  perpendicular  respectively  to  A  H,  A  H' ;  hence 

C  A  C  =  H  A  H'  =  A  H,  the  dip. 

Let     R  =  C  B,  the  radius  of  the  earth  ; 
then  R  +  h  =  C  A, 

7J?-CA.C  H",  the  radius  of  curvature  of  H"  A, 

6  R  =  C  C. 

We  have,  then,  in  the  triangle  C  A  C,  by  Pl.  Trio.  (268), 


f  V     7R(R+h)      ' 

and,  since  A  is  comparatively  very  small,  and  may  therefore 
be  omitted  alongside  of  R, 

I  O    I 

i   a  tt        a  ion 
sin  *  A  H  =  1/ ; 

*  \  7  R 

or,  putting  sin  J  A  ZT=  J  A  ZTsin  1", 

A^=sTnT>V7^=sin-rV7^V^  (32> 

46.    Taking  R  =  20902433  feet,  we  find  the  constant  factor 

2 


pVnr69''071' 


sin 

A#=59".071  y/h,  (33) 

and  log  A  IT=  1.77137  -f  }  log  h, 

h  being  expressed  in  feet,  which  is  nearly  the  formula  for 
Table  14  (Bowd.). 

2        /~3 
Since  - — — ,  y  — —  is  constant,  depending   only  upon  the 

radius  of  the  earth,  A  H  is  proportional  to  >/h,  or  the  dip 
is  proportional  to  the  square  root  of  the  height  of  the  ob- 
server above  the  level  of  the  sea. 


38  NAVIGATION. 


47.    Were  the  path  of  the  ray,  H"  A,  a  straight  line,  we 
>uld  have  \r  H  —  R  1 

and  in  the  triangle  H"  C  A, 


should  have         A'jy=HAH"=H"CA 


cosA'ZT= 


R  +  h' 

whence,  2  sin2  £  A'  H  =  — =  —  ,  nearly, 

R  -\-  h      R 

or  with  h  in  feet,  A'  H=  63".803  VA.  (34) 

Comparing  this  with        A  £T=  59".07  Vh,  we  find 

A  JI=  A'  #-  4".733  Vh  =  A'  ^T-  .074  A'  H, 
or  that  the  dip  is  decreased  by  refraction  by  .074,  or  nearly 

tV  of  it- 

But  from  the  irregularity  of  the  refraction  of  horizontal 
rays  (Art.  44),  the  dip  varies  considerably,  so  that  the  tabu- 
lated dip  for  the  height  of  16  feet  can  be  relied  on  ordinarily 
only  within  2'.  When  the  temperatures  of  the  air  and  water 
differ  greatly,  variations  of  the  dip  from  its  mean  value  as 
great  as  4'  may  be  experienced.  In  some  rare  cases,  varia- 
tions of  8'  have  been  found. 

The  dip  may  be  directly  measured  by  a  dip-sector.  A 
series  of  such  measurements  carefully  made,  and  under  dif- 
ferent circumstances,  both  as  to  the  height  of  the  eye,  tem- 
perature and  pressure  of  the  atmosphere,  and  temperature  of 
the  water,  is  greatly  needed. 

48.  Professor  Chauvenet  (Astron.,  1, 176)  has  deduced  the 
following  formula,  which  it  is  desirable  to  test  by  observa- 
tions .  — 


DIP   OF  THE  HORIZON.  39 


in  seconds,  A#  =  M H -  24021"  t 


a' a' 

or  in  minutes,  A  H=  A'  H-  6'.67  Lzih  ; 

A'^  ' 

in  which  t   is  the  temperature  of  the  air, 

t0  that  of  the  water, 
by  a  Fahrenheit  thermometer. 

When  the  sea  is  warmer  than  the  air,  the  visible  horizon 
is  found  to  be  below  its  mean  position,  or  the  dip  is  greater 
than  the  tabulated  value ;  when  the  sea  is  colder  than  the 
air,  the  dip  is  less  than  its  tabulated  value.  (Raper's  Nav., 
p.  61.) 

This  uncertainty  of  the  dip  affects  to  the  same  extent  all 
altitudes  observed  with  the  sea  horizon. 

49.  Near  the  shore,  or  in  a  harbor,  the  horizon  may  be 
obstructed  by  the  land.  (Bowd.,  Art.  253.)  The  shore-line 
may  then  be  used  for  altitudes  instead  of  the  proper  horizon. 
Table  15  (Bowd.)  contains  the  dip  of  such  water-line,  or  of 
any  object  on  the  water,  for  different  heights  in  feet  and  dis- 
tances in  sea  miles.     It  is  computed  by  the  formula 


in  which 


D  =  |^+0.56514^  (35) 

h  is  the  height  in  feet ; 
d,  the  distance  of  the  object  in  sea  miles ; 
D,  the  dip  in  minutes. 

50.    Problem  12.     To  find  the  distance  of  an  object  of 
known  height,  which  is  just  visible  in  the  horizon. 

Solution.     If  the  observer  is  at  the  surface  of  the  earth  at  the 
point  H"  (Fig.  15),  a  point  A  appears  in  the  horizon,  or  is  just 


40 


NAVIGATION. 


Tig,  14 


visible,  when  the  visual  ray  A  H" 
just  touches  the  earth  at  H".     Let 

h  =  B  A,  the  height  of  A, 
d  =  H"  A,  the  distance  of  A. 
As  this  arc  is  very  small,  we  have 
d=H"C'Asinl"  x  C'A 
=  7JRXH"C'A  sin  1", 
since  C  A  =  7  R. 

From  the  three  sides  of   the   tri- 
angle CC'Aby  Pl.  Trig.  (268), 


or  nearly      \  H"  C  A  sin  1"  =  \J~ , 


and 


H"  C  A  sin  V 


vO 


21  B 


This,  substituted  in  the  expression  for  d,  gives 


d=1R\hE=^iRh)  <36> 


In  this,  d,  h,  and  It,  are  expressed  in  the  same  denomination. 
But  if  h  and  H  are  in  feet, 

in  statute  miles,  d 


5280 


in  geographical  miles, 


Taking  i?  ==  20902433  feet  as  before,  we  find 
in  stat.  miles  d  =  1.323  VA,  or  log  d  =  0.12156  +  \  log  h,  | 
in  geog.     "    d  =  1.148  Vh,  or  log  d  =  0.05994  +  J  log  A.  J  ^ 

The  first  of  these  is  nearly  the  formula  given  in  Bowditch 
for  computing  Table  6. 


PARALLAX. 


41 


51.    Were  the  visual  ray,  H"  A,  a  straight  line,  we  should 
have  from  the  right  triangle  C  H"  A, 


H"  A  =  V  (c  A2  ~  H"  c*)>  or  d'  =  V  (2X  +  A)  A ; 

or  nearly  oT  =  VOJ  X  VA. 

Introducing  the  same  numerical  values  as  before,  we  have 
in  statute  miles 

d'  =  1.225  VA. 

Comparing  this  with  the  expression  above,  we  see  that  the 
distance  is  increased  about  T\  part  by  refraction.  This,  how- 
ever, is  subject  to  great  uncertainty. 


52.  If  the  observer  is  also  elevated  at  the  height  of  B'  A' 
(Fig.  16),  and  sees  the  object  A  in  his  horizon,  then  its  dis- 

ta"0e  iS  A'  H"  +  H"  A, 

or  the   sum    of  the  distances  of  each 
from  the  common  horizon,  H". 

By  entering  Table  6  with  the  heights 
of  the  observer  and  the  object  respect- 
ively,   the    sum    of    the    corresponding 
distances  is  the  distance  of  the  object 
from   the   observer.     The   distances  in 
^98n  this  table  are  in  statute  miles.     Multi- 
plying them  by  6qqa~o  =  .86839,  reduces  them  to  geographical 
miles. 

PARALLAX. 

53.  The  change  of  the  direction  of  an  object,  arising  from 
a  change  of  the  point  from  which  it  is  viewed,  is  called  paral- 
lax ;  and  it  is  always  expressed  by  the  angle  at  the  object, 
which  is  subtended  by  the  line  joining  the  two  points  of  view. 


42 


NAVIGATION. 


Thus  in  Fig.  17,  the  object  S  would  be  seen  from  A  in  the 
direction  A  S ;  and  from  C  in  the  direction  C  S.  The  angle 
at  S,  subtended  by  A  C,  is  the  difference  of  these  directions, 
or  the  parallax  for  the  two  points  of  view,  C  and  A. 

54.  In  astronomical  ob- 
servations, the  observer  is 
on  the  surface  of  the  earth ; 
the  conventional  point  to 
which  it  is  most  convenient 
to  reduce  them,  wherever 
they  may  be  made,  is  the 
earth's  centre.  In  those 
problems  of  practical  as- 
tronomy which  are  used  by 
the  navigator,  we  have  only 
to  consider  this  geocentric 
parallax,  which  is  the  dif- 
ference of  the  direction  of  a  body  seen  from  the  surface  •  and 
from  the  centre  of  the  earth.  It  may  also  be  defined  to  be 
the  angle  at  the  body  subtended  by  that  radius  of  the  earth 
which  passes  through  the  place  of  the  observer.     Thus,  in 

Fig.  17,  if 

C  is  the  centre  of  the  earth,  and 

A  the  place  of  the  observer, 

the  geocentric  parallax  of  a  body,  S,  will  be  the  angle 

S  =  ZAS-ZCS, 

at  the  body  subtended  by  the  radius  C  A. 

If  the  earth  is  regarded  as  a  sphere,  C  A  Z  will  be  the 
vertical  line  through  A,  and  will  pass  through  the  zenith,  Z. 
Then  will  the  plane  of  C  A  S  be  a  vertical  plane ; 


Fig.  17. 


PARALLAX.  43 

ZAS,  the  apparent  zenith  distance  of  S  as  observed  at  A ; 
ZCS,  its  geocentric  or  true  zenith  distance ;  and 
ZAS>  ZCS. 

Thus  we  see  that  this  parallax  takes  place  in  a  vertical  plane, 
and  increases  the  zenith  distance,  or  decreases  the  altitude,  of 
a  heavenly  body  without  affecting  its  azimuth. 

55.  This  suffices  for  all  nautical  problems  except  the  com- 
plete reduction  of  lunar  distances. 
For  these  and  the  more  refined  obser- 
vations at  observatories,  the  spheroidal 
form  of  the  earth  must  be  considered. 
Then,  as  in  Fig.  18,  the  radius  C  A 
does  not  coincide  with  the  normal  or 
vertical  line  C  A  Z,  but  meets  the 
celestial   sphere  at  a  point  Z',  in   the 

celestial  meridian,  nearer  the  equator  than  the  zenith,  Z. 

We  may  remark  here  that 
A  C"  E,  the  angle  which  the   vertical   line    makes  with  the 

equator,  is  the  latitude  of  A  ;  and 
ACE,  the  angle  which  the  radius  makes  with  the  equator  is 

its  geocentric  latitude. 

56.  Problem  13.     To  find  the  parallax  of  a  heavenly- 
body  for  a  given  altitude. 

Solution.     In  Fig.  17  let 
p  =  S,  the  parallax  in  altitude ; 
z  =  Z  A  S,  the  apparent  zenith  distance  of  S,  corrected  for 

refraction ; 
R  =  C  A,  the  radius  of  the  earth ; 

c?=CS,  the  distance  of  the  body  S,  from  the  centre  of  the 
earth. 


44  NAVIGATION. 

Then  from  the  triangle  C  A  S,  we  have 

C  A 

sin  C  S  A  =  ^-^  sin  CAS, 

B  sin  z 
or,  sm^= — — ,  (38) 

If  the  object  is  in  the  horizon  as  at  H,  the  angle  A  H  C  is 
called  its  horizontal  parallax ;  and  denoting  it  by  P,  we  have 
from  (38),  or  from  the  right  triangle  C  A  H, 

sinP  =  — ,  (39) 

which,  substituted  in  (38),  gives 

sin  p  =  sin  P  sin  z.  (40) 

If   h  =  90°  —  z,  the  apparent  altitude  of  the  object,  we 

have  — 

sin  p  =  sin  P  cos  h ;  (41) 

or  nearly,  since  p  and  P  are  small  angles, 

p  —  P  cos  h.  (42) 

57.  The  horizontal  parallax  P,  is  given  in  the  Nautical 
Almanac  for  the  sun,  moon,  and  planets.  From  Fig.  17  it  is 
obviously  the  semidiameter  of  the  earth,  as  viewed  from  the 
body.  As  the  equatorial  semidiameter  is  larger  than  any 
other,  so  also  will  be  the  equatorial  horizontal  parallax.  This 
is  what  is  given  in  the  Almanac  for  the  moon.  Strictly,  it 
requires  reduction  for  the  latitude  of  the  observer,  and  such 
reduction  is  made  at  observatories,  and  in  the  higher  order  of 
astronomical  observations.     It  is  given  in  Table  19  (Bowd.). 

58.  Tables  16  and  17  (Bowd.)  are  computed  by  formula 
(42). 

Table  23  contains  the  correction  of  the  moon's  altitude  for 


APPARENT  SEMIDIAMETEBS. 


45 


parallax  and  refraction  corresponding  to  a  mean  value  of  the 
horizontal  parallax,  57'  30".  It  should  be  used,  however, 
only  for  very  rough  observations,  or  a  coarse  approximation. 

59.  Table  24  contains,  to  each  minute  of  horizontal  paral- 
lax, and  every  10'  of  altitude  from  5°,  the  combined  correction 
for  parallax  and  refraction  of  the  apparent  altitude  of  the 
moon's  centre  :  barom.,  30" ;  therm.,  50°  F.  Before  using 
this  table,  the  observed  altitude  of  the  moon's  limb  should 
be  corrected  for  instrumental  errors,  dip,  and  semidiameter. 


APPARENT    SEMIDIAMEIEEB. 

60.    The  apparent  diameter  of  a  body  is  the  angle  which 
its  disk  subtends  at  the  place  of  the  observer. 

Problem  14.      To  find  the  apparent  semidiameter  of  a 
heavenly  body. 

Solution.     In  Fig.  19,  let  M  be  the  body  ; 

d  =  C  M,  its  distance  from  the  centre  of  the  earth ; 

d!  =  A  M,  its  distance  from  A ; 

r  =  M  B,  its  linear  radius  or 
semidiameter  ; 

s  =  M  C  B,  its  apparent  semidi- 
ameter, as  viewed  from 
C; 

s'  =  M  A  B',  its  apparent  semi- 
diameter, as  viewed  from 
A  (B  and  B'  are  too  near 
each  other  to  be  distin- 
guished in  the  diagram) ; 

R  =  C  A,  the  earth's  radius. 


Fisr.  19. 


46  NAVIGATION. 

1.    For  finding  s,  the  right  triangle  C  B  M,  gives 


sin*  =  -.  (43) 


Were  the  body  M  in  the  horizon  of  A,  or  Z  A  M  =  90°,  its 
distance  from  A  and  C  would  be  sensibly  the  same,  so  that 
the  angle  s  is  called  the  horizontal  semidiameter. 

In  (39)  we  have  for  the  horizontal  parallax, 

.     "        B  R 

sin  P  —  — ,  or(/  = 


a  sin? 


which,  substituted  in  (43),  gives 


r 
sin  s  =  —  sin  P}  (44) 


R 

or  nearly,  since  s  and  P  are  small, 

*  =  ^P,  (45) 

T 

—  is  constant  for  any  particular  body,  as  it  is  simply  the 

ratio  of  its  linear  diameter  to  that  of  the  earth. 
For  the  moon, 

^=0.272, 

5=  0.272  P,  ) 

and  log  s  =  9.43457  +  log  P.  ) 

By  this  formula  the  moon's  horizontal  semidiameter  may 
be  found  from  its  horizontal  parallax.  (Naut.  Alm.,  p.  506.) 

The  Nautical  Almanac  contains  the  semidiameters  as  well 
as  the  horizontal  parallaxes  of  the  sun,  moon,  and  planets. 

2.  For  finding  /,  the  apparent  semidiameter  as  viewed 
by  an  observer  at  A  on  the  surface  of  the  earth,  the  right  tri- 
angle AB'M  gives  r 

sin  •  as  ^ .  (47) 


APPARENT  SEM1DIAMETERS.  47 


In  the  triangle  C  M  A, 

sin  M  A  C       CM 


sin  M  C  A       AM' 


or,  putting         h  =  90°  —  Z  A  M,  the  apparent, 

and  ti  =  90°  —  Z  C  M,  the  true  altitude  of  M, 

cos  h  _  d 
cos  ti       d 

-,  7/       j  cos  ti 

whence,  a  —  d , 

cos  h 

which,  substituted  in  (47),  and  by  (43),  gives 

,       r  cos  h  cos  h 

sin  6'  = ■  =  sin  s 


(48) 


dcosh'  cos  h'9 

cos  h 
cos  h! 


,     i  /  COS   «  ,,-v 

or  approximately,  s  =  s ,  (49) 


by  which  /  may  be  found  when  s  and  h  are  known. 

Since  h  <  N,  cos  A  >  cos  ti,  and  consequently  s'  >  s ;  that 

is,  the  semidiameter  increases  with  the  altitude  of  the  body. 

The  excess 

A  s  =  /  —  s,  is  called  the  augmentation. 

The  moon  is  the  only  body  for  which  this  augmentation  is 
sensible.     It  is  given  in  Table  18  (Bowd.). 


48  NAVIGATION. 


CHAPTER   III. 

TIME. 

61.  Transit.  The  instant  when  any  point  of  the  celestial 
sphere  is  on  a  given  meridian  is  designated  as  the  transit  of 
the  point  over  that  meridian. 

62.  Hour-angle.  The  hour-angle  of  any  point  of  the  sphere 
is  the  angle  at  the  pole  which  the  circle  of  declination  pass- 
ing through  the  point  makes  with  the  meridian.  It  is  prop- 
erly reckoned  from  the  upper  branch  of  the  meridian,  and 
positively  toward  the  west.  It  is  usually  expressed  in  hours, 
minutes,  and  seconds  of  time.  The  intercepted  arc  of  the 
equator  is  the  measure  of  this  angle. 

63.  Sidereal  Time.  The  intervals  between  the  successive 
transits  of  any  fixed  point  of  the  sphere  (as,  for  instance,  of 
a  star  which  has  no  proper  motion)  over  the  same  meridian 
would  be  perfectly  equal,  were  it  not  for  the  variable  effect 
of  nutation.  This  correction,  arising  from  a  change  in  the 
position  of  the  earth's  axis,  is  most  perceptible  in  its  effect 
upon  the  transit  of  stars  near  the  vanishing  point  of  that 
axis,  i.e.,  near  the  poles  of  the  heavens.  Hence,  for  the  exact 
measurement  of  time,  we  use  the  transits  of  some  point  of  the 
equator,  as  the  vernal  equinox.  This  point  is  often  called  the 
first  point  of  Aries.     Its  usual  symbol  is  °f . 


TIME.  49 

64.  The  interval  between  two  successive  transits  of  the 
vernal  equinox  is  a  sidereal  day  ;  and  such  a  day  is  regarded 
as  commencing  at  the  instant  of  the  transit  of  that  point. 
The  sidereal  time  is  then  0h  0m  0s.  This  instant  is  sometimes 
called  sidereal  noon. 

The  effect  of  nutation  and  precession  in  changing  the  time 
of  the  transit  of  the  vernal  equinox  is  so  nearly  the  same  at 
two  successive  transits,  that  the  sidereal  days  thus  denned 
are  sensibly  equal.  It  is  unnecessary,  then,  except  in  refined 
discussions,  to  discriminate  between  mean  and  apparent 
sidereal  time. 

65.  The  sidereal  time  at  any  instant  is  the  hour-angle  of 
the  vernal  equinox  at  that  instant,  and  is  reckoned  on  the 
equator  from  the  meridian  westward  around  the  entire  cir- 
cle ;  that  is,  from  0  to  24A  It  is  equal  to  the  right  ascension 
of  the  meridian  at  the  same  instant. 

66.  Solar  Time.  The  interval  between  two  successive 
transits  of  the  sun  over  a  given  meridian  is  a  solar  day,  and 
the  hour-angle  of  the  sun  at  any  instant  is  the  solar  time  of 
that  instant. 

In  consequence  of  the  motion  of  the  earth  about  the  sun 
from  west  to  east,  the  sun  appears  to  have  a  like  motion 
among  the  stars  at  such  a  rate  that  it  increases  its  right 
ascension  daily  nearly  1°,  or  4TO  of  time.  With  reference  to 
the  fixed  stars,  it  therefore  arrives  at  the  meridian  each  day 
about  4OT  later  than  on  the  previous  day ;  consequently,  solar 
days  are  about  4m  longer  than  sidereal  days. 

67.  Apparent  and  Mean  Solar  Time.  If  the  sun  changed 
its  right  ascension  uniformly  each  day,  solar  days  would  be 
exactly  equal.     But  the  sun's  motion  in  right  ascension  is  not 


50  NAVIGATION. 

uniform,  varying  from  Sm  35s  to  4m  265  in  a  solar  day.     There 
are  two  reasons  for  this,  — 

1.  The  sun  does  not  move  in  the  equator,  but  in  the 
ecliptic. 

2.  Its  motion  in  the  ecliptic  is  not  uniform,  being  most 
rapid  at  the  time  of  the  earth's  perihelion,  about  January  1, 
and  slowest  at  the  time  of  the  aphelion,  about  July  2. 

To  obtain  a  uniform  measure  of  time  depending  on  the 
sun's  motion,  the  following  method  is  adopted.  A  fictitious 
sun,  called  a  mean  sun,  is  supposed  to  move  uniformly  in  the 
ecliptic  at  such  a  rate  as  to  return  to  the  perigee  and  apogee 
at  the  same  time  with  the  true  sun.  A  second  mean  sun  is 
also  supposed  to  move  uniformly  in  the  equator  at  the  same 
rate  that  the  first  moves  in  the  ecliptic,  and  to  return  to  each 
equinox  at  the  same  time  with  the  first  mean  sun. 

The  time  which  is  measured  by  the  motion  of  this  sec- 
ond mean  sun  is  uniform  in  its  increase,  and  is  called  mean 
time. 

That  which  is  denoted  by  the  true  sun  is  called  true  or 
apparent  time. 

The  difference  between  mean  and  apparent  time  is  called 
the  equation  of  time.  It  is  also  the  difference  of  the  right 
ascensions  of  the  true  and  mean  suns. 

The  instant  of  transit  of  the  true  sun  over  a  given  merid- 
ian is  called  apparent  noon.  .  The  instant  of  transit  of  the 
second  mean  sun  is  called  mean  noon.  The  mean  time  is 
then  0h  0m  0s. 

Mean  noon  occurs,  then,  sometimes  before  and  sometimes 
after  apparent  noon,  the  greatest  difference  being  about  16m, 
early  in  November. 

68.    Astronomical    Time.     The    solar    day    (apparent   or 


TIME.  51 

mean)  is  regarded  by  astronomers  as  commencing  at  noon 
(apparent  or  mean),  and  is  divided  into  24  hours,  numbered 
successively  from  0  to  24. 

.Astronomical  time  (apparent  or  mean)  is,  then,  the  hour- 
angle  of  the  sun  (true  or  mean)  reckoned  on  the  equator 
westward  throughout  the  entire  circle  from  0*  to  24*. 

69.  Civil  Time.  For  the  common  purposes  of  life,  it  is 
more  convenient  to  begin  the  day  at  midnight ;  that  is,  when 
the  sun  is  on  the  meridian  below  the  horizon,  or  at  the  sun's 
lower  transit.  The  civil  day  begins  12*  before  the  astronomi- 
cal day  of  the  same  date ;  and  is  divided  into  two  periods  of 
12*  each,  namely,  from  midnight  to  noon,  marked  a.m.  (ante- 
meridian), and  from  noon  to  midnight,  marked  p.m.  (post- 
meridian).    Both  apparent  and  mean  time  are  used. 

The  affixes  a.m.  and  p.m.  distinguish  civil  time  from  astro- 
nomical time.  During  the  p.m.  period,  this  is  the  only  distinc- 
tion, —  the  day,  hours,  etc.,  being  the  same  in  both. 

70.  Sea- Time.  Formerly,  in  sea-usage,  the  day  was  sup- 
posed to  commence  at  noon,  12*  before  the  civil  day,  and  24* 
before  the  astronomical  day  of  the  same  date ;  and  was  divided 
into  two  periods,  the  same  as  the  civil  day.  Sea-time  is  now 
rarely  used. 

71.  To  convert  civil  into  astronomical  time,  it  is  only  neces- 
sary to  drop  the  a.m.  or  p.m.,  and  when  the  civil  time  is  a.m., 
deduct  ld  from  the  day,  and  increase  the  hours  by  12*. 

To  convert  astronomical  into  civil  time,  if  the  hours  are 
less  than  12*,  simply  affix  p.m.  ;  if  the  hours  are  12*  or  more 
than  12*,  deduct  12*,  add  ld,  and  affix  a.m. 


52 


NAVIGATION. 

Examples. 

1IC 

AL  TIME.                                     CIVIL  TIME. 

d 

h     m       a                                       d        h     m       8 

1860  May  10  14  15  10  =  1860  May  11    2  15  10  a.m. 

1862  Sept.  8    9  19  20  =  1862  Sept.  8    9  19  20  p.m. 

1863  Jan.    3  23  22  16  =  1863  Jan.    4  11  22  16  a.m. 
1868  Jan.    4    0    3  30  =  1863  Jan.    4    0    3  30  p.m. 

72.  The  hour-angle  of  the  sun  (true  or  mean),  at  any  me- 
ridian, is  called  the  local  (apparent  or  mean)  solar  time.  The 
hour-angle  of  the  sun  (true  or  mean)  at  Greenwich  at  the 
same  instant  is  the  corresponding   Greenwich  time. 

So  also  the  hour-angle  of  *f  at  any  meridian,  and  its  hour- 
angle  at  Greenwich  at  the  same  instant,  are  corresponding 
local  and  Greenwich  sidereal  times. 

73.  The  difference  of  the  local  times  of  any  two 
meridians  is  equal  to  the  difference  of  longitude  of  those 
meridians. 

Demonstration.     In  Fig.  20,  let 
PM,PM'  be  the  celestial  meridians 

of  two  places ; 
P  S,  the   declination  circle  through 

the  sun   (true  or  mean) ; 
MPS,  the  hour-angle  of  the  sun  at 

all  places  whose  meridian  is  P  M, 

will  be  the  local  time  (apparent  or  mean)  at  those  places ; 

so  also 
M'PS  will  be  the  corresponding  local   time    at   all   places 

whose  meridian  is  P  M' ;  and 
MPM'  =  MPS-M'PS  will  be  the  difference  of  longitude 

of  the  two  meridians. 

If  P  V  is  the  equinoctial  colure, 


TIME.  53 

MPV  and  M'PT  will  be  the  corresponding  sidereal  times  at 

the  two  meridians ;  still,  however, 
MPM^MPT-M'Pf. 

The  proposition  is  true,  then,  whether  the  times  compared 
are  apparent,  mean,  or  sidereal 

The  difference  of  longitude  is  here  expressed  in  time.     It 
is  readily  reduced  to  arc  by  observing  that 

24*  =  360c 


1°    _  £m 

1*  =    15° 

-  or   - 

V   =4S 

1*  =    15' 

1"  =  tV 

1*  =   15" 

In  comparing  corresponding  times  of  different  meridians, 
the  most  easterly  meridian  is  that  at  which  the  time  is  great- 
est. 

74.  If  (Fig.  20)  P  M  is  the  meridian  of  Greenwich, 

M  P  S  is  the  Greenwich  solar  time,  and 
MPM'  the  longitude  of  the  meridian  P  M'. 

MPM'  =  MP8-M'PS; 
so  also      MPM'  =  MPT-M'PT; 

or,  the  longitude  of  any  meridian  is  equal  to  the  difference  be- 
tween the  local  time  of  that  meridian  and  the  corresponding 
Greenwich  time. 

75.  If  we  put 

T0  =  M  P  S,  the  Greenwich  time, 
T  =  M'P  S,  the  corresponding  local  time, 
X  =  M  P  M',  the  longitude  of  the  meridian,  P  M', 
we  have  \  =  T0  —  T, 

and  T0  =  T  +  \ 

in  which  \  is  +  for  west  longitudes,  and  T0  and  T  are  sup- 
posed to  be  reckoned  always  westward  from  their  respective 


r;\  ^ 


54  NAVIGATION. 

meridians  from  0h  to  24* ;  that  is,  rl\  and  T  are  the  astronom- 
ical times,  which  should  always  be  used  in  all  astronomical 
computations. 

76.  Usually  the  first  operation,  in  most  computations  of 
nautical  astronomy  is  to  convert  the  local  civil  time  into  the 
corresponding  astronomical  time  (Art.  71). 

The  Greenwich  time  should  never  be  otherwise  expressed 
than  astronomically.  On  this  account  it  would  be  convenient 
to  have  chronometers  intended  for  nautical  or  astronomical 
purposes  marked  from  0h  to  24*,  instead  of  0h  to  12A  as  is  now 
customary  with  sea-chronometers. 

77.  The  second  operation  often  required  is  to  convert  the 
local  astronomical  time  into  Greenwich  time.  For  this  we 
have  (50),  which  numerically  is 

t       77  |    J  +  when  the  longitude  is  west, 
■Iq  =  J.  -4-  K  \  . 

(  —  when  it  is  east, 

and,  in  words,  gives  the  following 

Kule.  Having  expressed  the  local  time  astronomically, 
add  the  longitude,  if  west ;  subtract  it,  if  east ;  the  result  is 
the  corresponding  Greenwich  time. 


TIME. 

Examples. 

1.    In  Long.   76°  32'  W.,  the  local  time  being  1898,  April 
lrf9A3m10s  a.m.,  what  is  the  Greenwich  time  ? 

Local  Ast.  T.  =  March  31*21*3™  10s 

Longitude  =  +568 

G.  T.  =  April      1      2  9    18 


TIME.  55 

2.  In  Long.  30°  E.,  the  local  time  being  March  20rf  6''  Sm 

a.m.,  what  is  the  G.  T.  ? 

Loc.  Ast.  T.  =  March  19d18A3m 
Long.  =  —      2  0 

G.  T.  =  March  19    16  3 

3.  In  Long.  105°  15'  E.,  the  local  time  being  August 
21^4'' 3™  p.m.,  what  is  the  G.  T.  ? 

Loc.  Ast.  T.  =  August  21<*  4A  3m 
Long.  =  —  7   1 

G.  T.  =  August  20  21  2 

By  reversing  this  process,  that  is,  by  subtracting  the  longi- 
tude if  taest,  or  adding  it  if  east,  we  may  reduce  the  Green- 
wich time  to  the  corresponding  local  time. 

When  observations  are  noted  by  a  chronometer  regulated 
to  Greenwich  time,  an  approximate  knowledge  of  the  longi- 
tude and  local  time  is  necessary  in  order  to  determine  whether 
the  chronometer  time  is  a.m.,  or  p.m.,  and  thus  fix  the  true 
Greenwich  date.  If  the  time  is  a.m.,  the  hours  must  be  in- 
creased by  12*. 

Examples. 

1.  In  Long.  5A  W.,  about  Sn  p.m.,  on  August  3d,  the  Green- 
wich chronometer  shows  8*  ll"1 75,  and  is  fast  of  G.  T.  6m  10*. 
What  is  the  Greenwich  time  ? 

Approx.  Loc.  T.  Aug.  SdSh  G.  Chro.  8h  llm  7s 

Long.  -f-  5  Correction       —    6    10 

Approx.  G.  T.      Aug.  3^8*  G.  T.Aug/  3d  8*   im  57s 

2.  In  Long.  10*  E.,  about  1*  a.m.,  on  December  7d,  the  G. 

Chro.  shows  3*  Um  13s.5,  and  is  fast  25m  18s.7.     Find  the  G.  T, 

Approx.  Loc.  T.  Dec.  6d13A  G.  Chro.  3M4™13S.5 

Long.  —  10  Correction       —  25    18s.  7 

Approx.  G.  T.      Dec.  6d   3*  G.  T.  Dec.  6<*  2A  48m  54*.8 


56  NAVIGATION. 

3.    In  Long.  9*  12m  W.,  about  2h  a.m.,  on  February  13rt, 

the  G.  Chro.  shows  11*  21 m  13A.3,  and  is  fast  30™  3(K3.     Find 

the  G.  T. 

Approx.  Loc.  T.  Feb.  12d  14A  0m         G.  Chro.  ll*27m138.3 

Long.  +     9  12  Correction  —    30   30*.3 

Approx.  G.  T.      Feb.  12d  23A  12"'  G.  T.  Feb.   12^22^56m43s.0 

The  operations  on  the  approximate  times  may  be  per- 
formed mentally. 

78.  Standard  Time.  By  this  system,  introduced  origi- 
nally for  the  convenience  of  railways  and  now  adopted  by  the 
United  States  and  other  countries,  the  civil  mean  time  of  cer- 
tain standard  meridians  is  used  throughout  the  adjacent  dis- 
tricts. The  standard  meridians  are  one  hour  (15°)  apart,  and 
those  in  use  in  North  America  are  the  60th,  75th,  90th,  105th 
and  120th  meridians  west  from  Greenwich;  the  times  are 
designated  respectively  Intercolonial,  Eastern,  Central,  Moun- 
tain, and  Pacific.  The  belts  of  territory  for  7^°  on  each  side 
of  a  standard  use  as  far  as  possible  the  time  of  that  meridian. 

To  reduce  Local  Mean  Time  to  Standard  Time.  If  the 
local  meridian  is  E.  of  the  standard,  subtract  the  difference 
of  longitude  between  the  two  meridians  from  the  1.  m.  t.,  and 
if  W.,  add  it. 


THE  NAUTICAL  ALMANAC. 


CHAPTER  IV. 

THE     NAUTICAL    ALMANAC. 

79.  The  American  Ephemeris  and  Nautical  Almanac  "  is 
divided  into  two  distinct  parts.  One  part  is  designed  for  the 
special  use  of  navigators,  and  is  adapted  to  the  meridian  of 
Greenwich.  The  other  is  suited  to  the  convenience  of  astron- 
omers, on  this  continent  particularly,  and  is  adapted  to  the 
meridian  of  Washington." 

80.  The  Nautical  part  of  this  Ephemeris  and  the  British 
Nautical  Almanac  give  at  regular  intervals  of  Greenwich  time 
the  apparent  right  ascensions  and  declinations  of  the  sun, 
moon,  planets,  and  principal  fixed  stars,  the  equation  of  time, 
the  horizontal  parallaxes  and  semidiameters  of  the  sun,  moon, 
and  planets,  and  other  quantities,  some  of  which  little  concern 
the  navigator,  but  are  needed  by  astronomers. 

81.  Before  we  can  find  the  value  of  any  of  these  quanti- 
ties for  a  given  local  time,  we  must  first  find  the  correspond- 
ing Greenwich  time  (Art.  77).  When  this  time  is  exactly  one 
of  the  instants  for  which  the  required  quantity  is  put  down 
in  the  Almanac,  it  is  only  necessary  to  transcribe  the  quan- 
tity as  it  is  there  given.  When,  as  is  mostly  the  case,  the 
time  falls  between  two  Almanac  dates,  the  required  quantity 
is  to   be    obtained  by  interpolation.     And  generally,  except 


58  NAVIGATION. 

when  great  precision  is  desired,  it  is  sufficient  to  use  first 
differences  only  ;  that  is,  regard  the  changes  of  the  quantity 
as  proportional  to  the  small  intervals  of  time  which  are  em- 
ployed. 

Thus,  for  a  day,  the  change  of  the  sun's  right  ascension 
may  be  regarded  as  uniform,  so  that  for  lh  it  is  J?  of  the 
daily  change ;  for  2h,  ^ ;  and  in  general  for  any  part  of  a  day 
it  will  be  the  same  part  of  the  daily  change. 
Generally,  then,  if 

Ad  represent  the  quantity  in  the  Almanac  for  a  date  preced- 
ing the  given  Greenwich  time ; 

A1?  its  change  in  the  time,  T-, 

t,  the  time  after  the  Almanac  date  for  which  the  value  of  the 
quantity  is  required,  expressed  in  the  same  unit  as  T,  and 

A,  the  required  value  ; 

we  have,  A-A.  +  1*.  (61) 

When  A0  is  increasing,  A:  has  the  same  sign  as  A0 ;  but 
when  A0  is  decreasing,  Ax  has  the  opposite  sign. 

82.  If  the  given  time  is  nearer  the  subsequent  than  the 
preceding  Almanac  date,  it  may  be  convenient  to  interpolate 
backward.  If,  then,  Ax  represent  the  quantity  in  the  Almanac 
for  a  subsequent  Greenwich  date,  and  t'  the  time  before  the 
Almanac  date,  we  have 

A  =  A-l^.  (52) 

83.  The  Almanac  contains  the  rate  of  change,  or  difference 
of  each  of  the  principal  quantities  for  some  unit  of  time. 
Thus,  in  the  Ephemeris  of  the  sun  and  planets,  the  "  Diff.  for 
lh,"  in  part  of  that  of  the  moon,  the  "  Diff.  for  lm,"  are  given. 


THE   NAUTICAL    ALMANAC.  59 


:  2 +;.::} 


If  t  or  t'  is  expressed  in  the  same  unit  of  time  as  that  for 
which  the  "Diff.,"  Ax,  is  given,  formulas.  (51)  and  (52)  become 

A  =  A0  +  t   A1? 

A 

Thus,  for  using  hourly  differences,  we  wish  the  hours, 
minutes,  etc.,  of  the  Greenwich  time  expressed  in  hours  and 
parts  of  an  hour ;  for  using  the  differences  for  lm,  we  wish 
the  minutes  and  seconds  of  Greenwich  time  expressed  in  min- 
utes and  parts  of  a  minute.  Decimal  parts  are  usually  most 
convenient,  though  some  computers  prefer  aliquot  parts. 

84.  The  quantities  in  the  Almanac,  as  commonly  in  other 
mathematical  tables,  are  approximate  numbers,  that  is,  each 
is  given  only  to  the  nearest  unit  of  the  lowest  retained  order ; 
and  no  refinement  of  interpolation  can  give  a  result  to  a 
higher  degree  of  precision.  In  interpolating,  more  than  one 
lower  order  in  any  case  is  superfluous.  Thus,  the  sun's  dec- 
lination is  given  to  the  nearest  0".l,  and  in  no  way  can  we 
by  interpolation  obtain  a  value  which  will  be  reliable  within 
a  narrower  limit. 

Moreover,  the  Greenwich  times  are  uncertain  to  a  greater 
or  less  extent ;  and  if  first  differences  only  are  used,  the  in- 
terpolated result  can  be  regarded  as  true  only  within  much 
wider  limits  than  the  approximation  of  the  Ephemeris. 

In  interpolating,  then,  it  is  well  to  consider  the  degree  of 
approximation  which  is  wanted  in  any  particular  case ;  and 
if  the  nearest  1',  or  10",  or  1"  suffices,  contract  the  interpo- 
lation so  as  to  retain  at  the  most  one  lower  order;  or  else, 
consider  the  degree  of  approximation  attainable  in  any  par- 
ticular case,  and  contract  the  work  so  as  to  retain  only  the 
reliable  figures.  All  lower  orders  are  superfluous,  and  are 
deceptive,  as  giving   the  appearance  of  a   higher  degree  of 


60  NA  VIGA  TION. 

accuracy  than  has  actually  been  obtained ;  as,  for  instance, 
using  tenths  and  hundredths  of  seconds,  when  the  data  will 
give  a  result  reliable  within  2'  or  3'  only. 

85.  Should  it  be  desirable  to  interpolate  more  accurately 
than  can  be  done  by  first  differences  alone,  the  reduction  for 
second  differences  may  be  introduced  by  a  simple  process. 

Let  A2  be  the  change  of  Ax  in  the  time  T'.  .  Then,  instead 
of  Ax,  as  found  in  the  Almanac  for  the  nearest  Greenwich 
date,  we  may  substitute 

Ai  +  ^|r,A2;  (54) 

that  is,  the  value  of  A: ,  interpolated  for  \  t,  or  to  the  middle 
instant  between  the  Almanac  date  and  the  given  time.  This 
is  simply  using  the  mean  rate  of  change  during  the  in- 
terval. 

If  Ax ,  is  a  "  Diff.  for  lh  "  given  for  the  Almanac  for  each 
day,  T  =  24*;  if  Aj  is  a  "Diff.  for  lm"  given  in  the  Al- 
manac for  each  hour,   T'  =  60m. 

The  interpolation  of  Ax  to  the  middle  instant  may  often 
be  performed  mentally. 

Example. 

If  the  sun's  right  ascension  for  1898,  Jan.  30,  8*  9"*  time 
be  required,  we  find  in  the  Almanac, 

for  Jan.  30  0*     Ax  =  10".244 

A2  =  _  0".035 
31  0*     A,  =  10".209 

and  by  interpolation  for  Jan.  30  4*,  the  middle  instant  be- 
tween Jan.  30  0*  and  Jan.  30  8*, 

Aj  =  10,,.244  -  0".006  =  10".238, 


THE  NAUTICAL  ALMANAC.  61 

which  is  the  mean   hourly  change  in   the   interval  from  0h 
to  8*. 

86.  Formula  (54),  however,  applies  only  to  an  Ephemeris 
where  the  differences  for  lh  or  for  lm,  which  are  designated 
by  A1}  are  given  for  the  same  instants  of  Greenwich  time 
as  the  functions,  A,  to  which  they  belong.*  For  instance, 
the  "Diff.  for  lh"  given  for  noon  Jan.  ld,  is  in  the  Ameri- 
can Ephemeris  the  change  per  hour  at  Jan.  ld  0*;  and  the 
same  in  the  British  Almanac. 

87.  Problem  15.  To  find  from  the  Almanac  a  required 
quantity  for  a  given  mean  time  at  a  given  place. 

Solution.  The  preceding  considerations  lead  to  the  fol- 
lowing rule  :  — 

1.  Express  the  given  mean  time  astronomically,  stating 
the  day  as  well  as  the  hours,  etc.,  and  reduce  it  to  Green- 
wich mean  time  by  adding  the  longitude,  if  west;  subtracting, 
if  east. 

2.  Take  from  the  Almanac  for  the  nearest  preceding  mean 
time  date  the  required  quantity  and  the  corresponding  "  Diff. 
for  lh"  or  "  Diff.  for  lm,"  noting  the  name  or  sign  of  each ; 
multiply  the  "Din0,  for  1A"  by  the  hours  and  parts  of  an 
hour,  or  the  "  Diff.  for  lm "  by  the  minutes  and  parts  of 
a  minute,  of  the  remaining  Greenwich  time ;  and  add  the 
product  algebraically. 

Or,  take  out  for  the  nearest  subsequent  date  the  required 
quantity  and  its  difference ;  multiply  the  "  Din0."  by  the 
hours  and  parts  of  an  hour,  or  the  minutes  and  parts  of  a 

*  The  u  Prop.  Logs,  of  Diff."  of  the  Lunar  Distances  are  given 
for  the  middle  instant. 


62  NAVIGATION. 

minute,  of  the  interval  from  the  given  Greenwich  date  to  the 
Almanac  date ;  and  subtract  the  product  algebraically. 

When  greater  precision  is  required,  interpolate  the  differ- 
ence to  the  middle  instant  between  the  given  Greenwich  date 
and  the  Almanac  date,  and  use  the  result  instead  of  the  dif- 
ference given  in  the  Almanac. 

This  rule  is  applicable  to  all  those  quantities  which  are 
given  at  regular  intervals  of  Greenwich  mean  time,  except  the 
moon's  meridian  passage  and  age  and  lunar  distances. 

For  the  "  Sidereal  Time  at  Greenwich  Mean  Noon,"  on 
p.  II  of  each  month,  the  "Diff.  for  1*"  is  9S.8565 ;  Table  3 
of  the  American  Ephemeris,  for  converting  a  mean  solar  into 
a  sidereal  interval,  may  be  used  for  the  interpolation. 

The  "  Mean  Time  of  Sidereal  0h"  on  p.  Ill,  is  given  at  in- 
tervals of  24*  of  sidereal  time.  The  "  Diff.  for  lh  »  is  —  95.8296 ; 
and  Table  2,  for  converting  a  sidereal  into  a  mean  solar  inter- 
val, may  be  used. 

88.  The  quantities  given  in  the  American  Ephemeris  for 
Washington  mean  time  may  be  interpolated  in  the  same  way, 
by  reducing  the  local  time  to  Washington  time  instead  of  to 
Greenwich  time. 

89.  The  apparent  places  of  the  fixed  stars  are  given  in 
the  British  Almanac  for  the  upper  transit  over  the  meridian 
of  Greenwich ;  in  the  American,  for  the  upper  transit  over 
the  Meridian  of  Washington.  In  the  latter,  the  Washington 
mean  time  is  given.  The  sidereal  time  at  either  place  for  the 
instant  of  transit  is  the  right  ascension  of  the  star  (Art.  65). 

Generally,  the  position  given  for  the  nearest  day  suffices. 
But  if  greater  precision  is  required,  it  is  necessary  to  reduce 
the  local  mean  time  to  the  sidereal  time  of  the  prime  merid- 
ian, and  interpolate  for  it. 


THE  NAUTICAL  ALMANAC.  63 

90.  In  the  following  examples  the  required  quantities  are 
taken  from  the  American  Ephemeris,  and  interpolated  to  the 
nearest  second  by  first  differences  (53),  and  to  the  highest 
precision  attainable  by  2d  differences  (54).  [Ordinarily,  in- 
terpolation to  the  nearest  second  by  (53)  suflices  for  the 
practical  purposes  of  navigation.] 

Examples. 

For  the  local  mean  time,  1898,  Jan.  30d  9h  14m  30s  a.m. 

in  Long.  163°  14'  W.,  find  the  following  quantities  from  the 

Nautical  Almanac :  — 

The  equation  of  time. 

O's  right  ascension,  j)'s  declination, 

O's  declination,  3)'s  horizontal  parallax, 


J)  's  right  ascension,  J)  's  semidiameter ; 

The  R.  ascension  and  declination  of  a  Scorpii  (Antares). 

Ast.  mean  time,  1865,  Jan.  29<*  21A  14m  30s 
Long.  +10   52    56 

G.  mean  time,  1865,     Jan.  30     8     7    26 

8     7.433 
8.1239 


1.    The  Equatio7i  of  Time  (Page  II). 

m     s  to     s  s 

Jan.  30,  0*.  13  34.14  +  0.387  13  34.14  +  0.387    \  =  -  .035 

8.124  .035 


3.!  24X4=-006 

-3.15         .04  +0.381  (at  4A) 

.01  8.124 

(  3.048 
038 


;.15  \    .( 


+  3.10 

Subtractive  from  mean  time,  13  37.24 

2.    The  O's  right  ascension  (Page  II). 


I: 


008 
001 


64  NAVIGATION. 


-  —  X 


Jan.  30,  0*  20  52  32.2  +  10.244  20  52  32.23  +  10.244    \  =  —  .035 

(-82. 
+  1  23.2  j     1.0  24 

I      .2  +  10.238  (at  4A) 

20  53  55.4  f  81.904 

+  123.17        «g 

20  53  55.40    ^      *041 

3.  The  O's  declination  (Page  II). 

Jan.  30,  -  17  34  04.0    +  4L42      -  17  34  04.0    +  41.42   A2  =  +  .77 

331'4  ^X4  =  +  .1S 

4.1  24 

+ 5  36'5   I        .8  41.55  (at  4*) 

.2  (  332.40 

-17  23  27.5  +5375J       4.16 

-17  28  26.44  ^        *1'7 

4.  The  1>  's  right  ascension  (Page  XII). 

h  m      8  h    m      s 

Jan.  30,  8\        3  23  30.8  +  2.1083     3  23  30.77  +  2.1083       A2  =  +  -0026 
7'433  ^x  3.7= +.0002 


( 14.8  -  60 

+  15.7  J      .8  +  2.1085  (at  3m.7) 

I      .'1  7.433 

3  23  46.5 


5.    The  3>'s  declination  (Page  XII). 


Jan.  30,  3A,    +  23  26  05.9  +  6*24      +23  26  05.9    +6.240    A2  =  -.109 

43.7  _^x3.7=-.007 

+  46.4    {2.5  60 


{43.1 


.2  +  6.233  (at  3m.7) 

+  23  26  52.3  c  43  63 

-46.3  J     2.40 
.02 


THE  NAUTICAL   ALMANAC.  65 

6.    The  D's  horizontal  parallax  (Page  IV). 


Jan.  30,  0*, 

54  24.7  -  0.78 

54  24.7  -  0.78        A2  =  +  0.21 

(  6.2 
-6.3   |     j 

•■|x4=+    .07 

54  18.4 

-0.71  (at  4*) 
i  5.68 
~5'8  |    .09 

54  18.9 

7.    TAe  J>  9g  semidiameter  (Page  IV). 


Jan.  30,  0* 

14  51.4  -  2.2  in  12* 

14  51.4 

5.77 

-1.5             in  8* 

.272 

14  49.9 

r  .115 

-1.6 

\  .040 
(-.001 

14  49.8 

In  Art.  (60)  we  have  for  the  moon,  s  =  .272  P\   whence 

As  =  .272  AP: 
so  that  the  reduction  of  the  semidiameter  may  be  readily 
found  by  multiplying  that  of  the  horizontal  parallax  by  .272, 
as  in  the  above  example.     (Naut.  Alm.,  p.  506.) 

The  right  ascension  and  declination  of  a  Scorpii  (Antares). 

The  Washington  (long.  +  5*  08m  12s)  mean  time  is  Jan.  30, 
2h  59m  14s,  or  Jan.  30.124.  On  page  299,  which  serves  as  an 
index,  the  mean  R.  A.  is  16*  23m.  The  apparent  R.  A.  and 
Dec.  (p.  346)  are  for  Jan.  29.8  m.  t.  Washington. 

h     m      s  s  o      /       //  // 

R.  A.  16  23  09.92  +  0.35  Dec.  -  26  12  24.6    -  0.7  (lOd) 

change  in  +  0.325d  +  .01  —  .02 

16  23  09.43  —  26  12  24.62 

91.  Problem  16.  To  find  from  the  Almanac  the  sun's 
right  ascension  and  declination,  and  the  equation  of  time 
for  a  given  apparent  time  at  a  given  place. 


66  NAVIGATION. 

Solution.  This  differs  from  the  preceding  problem  simply 
in  using  the  apparent  instead  of  the  mean  time,  and  in  taking 
the  quantities  from  page  I  for  the  month,  where  they  are 
given  for  apparent  noon,  instead  of  from  page  II,  where  they 
are  given  for  mean  noon. 

92.  Problem  17.  To  find  the  right  ascension  and  dec- 
lination of  the  sun,  and  the  equation  of  time  at  apparent 
noon  of  a  given  place,  or  when  the  sun  is  on  the  meridian. 

Solution.  The  local  apparent  time  is  0h  0m  0s.  The  Green- 
wich apparent  time  is  then  equal  to  the  longitude  if  west ;  that 
is,  it  is  after  the  noon  of  the  same  date  by  a  number  of  hours, 
etc.,  equal  to  the  longitude.  If  the  longitude  is  east,  the 
Greenwich  apparent  time  is  before  the  noon  of  the  same  date 
by  a  number  of  hours,  etc.,  equal  to  the  longitude. 

Hence,  take  these  quantities  from  the  Almanac  for  Green- 
wich apparent  noon  (p.  I)  of  the  same  day  as  the  local  (civil) 
day,  and  apply  a  correction  equal  to  the  hourly  difference  mul- 
tiplied by  the  hours  and  parts  of  an  hour  of  the  longitude ; 
observing  to  add  or  subtract  the  correction,  according  as  the 
numbers  in  the  Almanac  may  require,  for  a  time  after  noon, 
if  the  longitude  is  west ;  for  a  time  before  noon,  if  the  longi- 
tude is  east. 

Examples. 

1.  Mnd  the  sun's  right  ascension  and  declination,  and  the 
equation  of  time  for  apparent  noon,  1898,  Jan.  30,  in  Long. 
163°  14'  W. 

h     m      s  h      m      s 

1.    Long.  +  10  52  56         O  's  R.  A.  20  52  34.55  +    10.237  h 

=  +  10.882  f  102.37    in    10 

+  1  51.40  J      8'19    in        '8 
.82    in        .08 

20  54  25.95    I        .02    in        .002 


THE  NAUTICAL  ALMANAC.  67 

m      a  s 

O  's  dec.  -  17°  33'  54". 6  +  41".61  Eq.  of  t.  +  13  34.23  +  0.379 

''416'U  ^3.79 

+  r8srJ     33.29  +4.12        .30 

1      3  .33  I    .03 

.08 
-  17026'21".8  +13  38.35 


2.    For  apparent  hood,  1898,  March  21,  in  Long.  163°  14'  E. 

h      m      s  h    m    8  s 

2.    Long.  -  10  52  56  Q's  R.  A.  0  03  20.43  +  9.103 

=  —  10.882  f  91.03 

-139.06^    7'28 
.73 

I     .02 
0  01  41.37 


O's  dec.  +  0°21'44".8  +  59".23  Eq.  of  t.  +  7  13.44  -  0.752 

{592".3  (7.52 

47-38  +8.18         .60 

4".74  I  .06 
.12 

+  0°ir00//.3    .  +  7  21.62 

In  the  first  and  second  examples  the  Diffs.  for  lh  have 
been  interpolated  for  5^.5  or  half  the  longitude,  forward  in 
the  first,  back  in  the  second ;  ordinarily  such  precision  is  un- 
necessary. 

93.  Problem  18.  To  find  the  right  ascension  of  the 
mean  sun  for  a  given  time  and  place. 

Solution.  At  the  instant  of  mean  noon,  or  when  the  mean 
sun  is  on  the  meridian,  at  any  place,  the  right  ascension  of  the 
mean  sun  is  equal  to  the  sidereal  time.  The  quantity  on  page 
II  of  each  month,  in  the  Almanac,  called  "  sidereal  time/'  is 
also  the  right  ascension  of  the  mean  sun  at  Greenwich  mean 
noon,  and  may  be  interpolated  for  a  given  local  time  in  the 


68  NAVIGATION. 

same  way  as  the  right  ascension  of  the  true  sun.  (Pkob.  15.) 
The  constant  "  Diff.  for  lh  "  is  9S.8565.  A  table  for  convert- 
ing mean  time  into  sidereal  time  intervals  (Table  III)  facili- 
tates the  interpolation. 

We  have  also  the  right  ascension  of  the  mean  sun  equal  to 
that  of  the  true  sun  +  the  equation  of  time,  using  for  the 
equation  of  time  the  sign  of  its  application  to  mean  time. 

94.  Problem  19.  To  find  the  mean  time  of  the  moon's 
transit  over  a  given  meridian  on  a  given  day. 

Solution.  The  Almanac  contains  the  mean  time  of  each 
transit  of  the  moon  over  the  meridian  of  Greenwich  (p.  IV). 
This  mean  time  is  the  hour-angle  of  the  mean  sun  (Art.  72) 
when  the  moon  is  on  the  meridian ;  and  is  therefore  the  dif- 
ference of  right  ascension  of  the  moon  and  the  mean  sun.  As 
this  difference  is  constantly  increasing,  in  consequence  of  the 
moon's  more  rapid  increase  of  right  ascension,  the  mean  time 
of  each  transit  is  later  than  that  of  the  one  preceding  by  a 
number  of  minutes,  varying,  according  to  the  rate  of  the  moon's 
motion  from  40m  to  66m. 

If,  then,  Tx  and  T2  denote  the  mean  times  of  two  successive 
transits  of  the  moon  over  the  Greenwich  meridian,  T2  —  T1  is 
the  retardation  of  the  moon  in  passing  over  24J1  of  longitude ; 
so  that  for  any  longitude  A.  (expressed  in  hours)  the  retarda- 
tion is  nearly 

The  mean  time  of  a  transit  is,  then,  reduced  from  the 
Greenwich  to  any  other  meridian  by  interpolating  for  the 
longitude;  forward,  if  the  longitude  is  west;  backward,  if 
the  longitude  is  east,  since  east  longitudes  are  regarded  as 
negative. 


THE  NAUTICAL  ALMANAC.  69 

The  American  Ephemeris  gives  also  the  hourly  differences, 
which  facilitate  the  interpolation.  For  greater  exactness, 
these  differences  may  be  interpolated  for  half  the  longitude. 
The  practical  rule  will  be  :  — 

Take  from  the  Almanac  the  mean  time  of  meridian  passage 
for  the  given  astronomical  *  day,  and  add  to  it  the  product  of 
the  "  Diff.  for  lh  "  by  the  longitude  in  hours,  if  the  longitude 
is  west;  subtract  that  product  if  the  longitude  is  east;  or  it 
may  be  taken  from  Table  2  (Bowd.).  The  mean  time  of  merid- 
ian passage  for  the  given  day,  and  that  for  the  day  following 
in  west  longitude,  or  for  the  day  preceding  in  east  longitude, 
are  those  which  are  commonly  used.  But  it  is  more  exact  to 
use  half  the  difference  of  the  times  of  meridian  passage  for 
the  day  preceding  and  the  day  following  the  given  day  :  ^¥  of 
this  is  the  "  Diff.  for  lh  "  of  the  American  Ephemeris. 

The  times  of  transit  are  given  only  to  tenths  of  a  minute, 
which  suffices  the  purposes  of  the  navigator.  They  may  be 
found  more  exactly  for  any  meridian  by  the  method  here- 
after given  in  Problem  27. 

95.  Problem  20.  To  find  on  a  given  day  the  mean 
time  of  transit  of  a  planet  over  a  given  meridian. 

Solution.  The  mean  time  of  each  meridian  passage  at 
Greenwich  is  given,  in  the  Almanac,  for  each  planet.  It 
may  be  reduced  to  any  meridian  in  the  same  way  as  for  the 
moon ;  except  that,  in  the  case  of  an  acceleration,  the  sign  of 
the  reduction  is  reversed. 

*  It  is  important  to  notice  whether  the  mean  time  of  transit  is  more 
or  less  than  12A.  In  the  former  case,  the  astronomical  day  is  ld  less 
than  the  civil  day. 


70  NAVIGATION. 

Examples. 

1.    In  Long.  100°  15'  W.,  find  the  times  of  meridian  passage 
of  the  moon  and  Jupiter  for  1898,  June  7  (civil  day). 

Long.  +  6*41-0'  =  +  6\683. 

J>  % 

h      71%  Wl  H     ftl  771 

M.  T.  of  mer.  pass.,  June  6, 14  32.2  +  2^49   June  7,  6  59.4  —3^  in  1  d. 

114.94  r  0.95  in  6*  . 

1.49  -1.1    \    .10  in  .6 

.20  I    .01  in  .08 

.01 
June  7,     2  48.8,  a.m.  June  7,     6  58.3,  p.m. 


2.  In  Long.  100°  15'  E.,  for  1898,  June  7  (civil  day),  find 
the  times  of  meridian  passage  for  the  moon  and  Jupiter. 

Long.  -6*41-0*  =  -  6A.683. 
D  % 

h     m  to  Am  m 

M.  T.  of  mer.  pass.,  June  6, 14  32.2  +  2^55    June  7,  6  59.4  -3.8   in  1  d. 

{15.30  r  0.95  in  Qh 

1.53  +  1-1    \     .10  in  .6 

.20  I    .01  in  .08 

.01 
June  7,     2  15.2,  a.m.  June  7,     7  00.5,  p.m. 

In  the  case  of  the  moon  the  hourly  differences  have  been 
interpolated  for  half  the  longitude.  (Ordinarily  this  pre- 
cision is  unnecessary.) 

96.  Problem  21.  To  find  the  right  ascension  or  dec- 
lination of  the  moon,  or  a  planet,  at  the  time  of  its 
transit  over  a  given  meridian  on  a  given  day. 

Solution.  Find  the  local  mean  time  of  transit,  as  in  Prob- 
lem 19;  deduce  the  corresponding  Greenwich  time  by  ap- 
plying the  longitude ;  and  for  this  Greenwich  time  take  out 
the  right  ascension  or  declination,  as  in  Problem  15. 


THE  NAUTICAL   ALMANAC.  71 

If  the  time  of  transit  has  been  noted  by  a  clock  or  chro- 
nometer, regulated  to  either  local  or  Greenwich  time,  it  should 
be  used  in  preference  to  the  time  of  transit  computed  from 
the  Almanac. 

97.  Problem  22.  To  find  the  Greenwich  mean  time 
of  a  given  lunar  distance. 

Solution.  The  angular  distances  of  the  moon  from  the  sun, 
the  principal  planets,  and  several  selected  stars,  are  given  in 
the  Almanac  for  each  Sh  of  Greenwich  mean  time. 

If  d  represent  the  given  distance  ; 

d0,  the  nearest  distance  of  the  same  body  in  the  Almanac 

preceding  in  time  the  given  distance ; 
Aj. ,  the  change  of  distance  in  3A ; 
t,  the  required  time  (in  hours)  from  the  date  of  d0 ; 

by  (51)  we  have  approximately,  using  1st  differences  only, 
whence,  for  the  inverse  interpolation, 

or,  with  t  in  seconds  of  time,  which  is  better  for  computation, 

t  =  —£—(d-dQ),  (57) 

in  which  it  is  most  convenient  to  express  Ax  and  (d  —  d0)  in 
seconds. 

Then  by  logarithms  : 

log  t  =  log  (d  -  cl,)  +  log  ^00  f  (58) 


10800 
ence  log 

ar.  complement  of  the  "  log  diff.  for  Is." 


S— .  is  the  change  of  distance  in  1* ;  hence  log is  the 

10800  8  Ax 


72  NAVIGATION. 

It  is  given  in  the  Almanac  for  the  middle  instant  between 
the  tabulated  distances  under  the  head  "  P.  L.*  of  Diff." ; 
the  index,  which  is  0,  and  the  separatrix  being  omitted. 

In  the  same  way,  if 

di  represent  the  distance  in  the  Almanac  following  the  given 

distance  ;  and 
tf,  the  interval  before  the  date  of  dlf 

we  shall  have  by  (52) 

and  tf  =  —  (di  -  d), 

or  with  if  in  seconds,  and  by  logarithms, 

log  t'=  log  (<£  -  d)  +  log  i^OO  _  (59) 

The  computation  is  simplified  by  using  a  table  of  "  loga- 
rithms of  small  arcs  in  space  or  time."  f  It  differs  from  the 
common  table  of  logarithms  only  in  having  the  argument  in 
sexagesimal  instead  of  natural  numbers.  With  such  a  table 
it  is  unnecessary  to  reduce  differences  of  distance  to  seconds, 
or  to  first  find  the  intervals  of  time  in  seconds. 

From  (58)  and  (59)  we  have  the  following  rule  :  Find  in 
the  Almanac  the  two  distances  between  which  the  given 
distance  falls ;  take  out  the  nearest  of  these,  the  hours  of 
Greenwich  time  over  it,  and  the  "  P.  L.  of  Diff."  between 
them.  Find  the  difference  between  the  distance  taken  from 
the  Almanac  and  the  given  distance ;  and  to  the  log.  of  this 
difference  add  the  "  P.  L.  of  Diff."  from  the  Almanac.  The 
sum  is  the  log.  of  an  interval  of  time  to  be  added  to  the  hours 
of  Greenwich  time  taken  from  the  Almanac,  when  the  earlier 
*  Proportional  Logarithm.  t  Table  34  (Bowd.). 


THE  NAUTICAL  ALMANAC.  73 

Almanac  distance  is  used ;  to  be  subtracted  from  the  hours  of 
Greenwich  time  when  the  later  Almanac  distance  is  used. 
(Chauvenet's  "  Lunar  Method/'  p.  8.) 

98.  The  result,  however,  may  not  be  sufficiently  approxi- 
mate, owing  to  the  neglect  of  2d  differences.  To  correct  it  for 
2d  differences,  Table  10  of  Chauvenet's  Method,  Table  I  of 
the  Almanac,  or  Table  35  (Bowd.)  may  be  used.  For  either, 
take  the  difference  between  the  two  Prop.  Logs.,  which  pre- 
cede and  follow  the  one  taken  from  the  Almanac.  With  half 
this  difference,  and  the  interval  of  time  just  found,  enter  the 
table  and  take  out  the  seconds,  which  are  to  be  added  to 
the  approximate  Greenwich  time  when  the  Prop.  Logs,  are 
decreasing,  but  subtracted  when  they  are  increasing. 

Second  differences  may  also  be  introduced  by  first  finding, 
or  estimating,  the  Greenwich  mean  time  to  the  nearest  10m, 
and  interpolating  the  Prop.  Log.  in  the  Almanac  to  the  middle 
instant  between  that  time  and  the  Almanac  hour  used,  as  in 
Art.  88  for  direct  interpolation. 

99.  Maskelyne,  the  author  of  the  present  arrangement  of 
lunar  distances,  to  facilitate  their  interpolation,  devised  what 
he  chose  to  call  proportional  logarithms. 

If  n  represent  any  number  of  seconds,  either  of  space  or 

time,  the  proportional  logarithm  of  n  is  the  log.  of 

Table  45  (Bowd.)  contains  these  proportional  logarithms 
for  each  second  of  n  from  0  to  3°,  or  to  3^,  the  argument  being 
in  °  '  "  or  in  h  m  s.  But  such  a  table  is  less  useful  for  other 
purposes  than  Table  I  of  the  American  Ephemeris,  previously 
referred  to. 

Dividing  both  members  of  (57)  by  10800,  and  inverting, 
we  have 


74  NAVIGATION. 


10800  _      Ax  10800 


«  10800       tf  -  <% ' 

and  P.  log  £  =  P.  log  (d  —  c?0)  —  P.  log  Ax , 

which  accords  with  the  rule  in  Art.  310  (Bowd.). 

100.    Example. 

1898,  Oct.  27,  the  distance  of  Fomalhaut  from  the  moon's 
centre  is  52°  3'  35",  what  is  the  Greenwich  mean  time  ? 

d  =    52    3  35 
Oct.  27,  15*,    d0  =    51  41  15  P.  log  0.3323  diff.  —  22 

d  —  d0=         22  20  log  3.1271 

t    =  +  0  48  00  log  3.4594 

Red  for  2d  diff.  +  05  Table  35  (Bowd.). 

G.  m.  t.,  Oct.  27,       15  48  05 

or,  by  back  interpolation, 

d  =    52    3  35 
Oct.  27,  18*,    dx  =    53    5  P.  log  0.3323  diff.  —  22 

d1  —  d=      1    125  log  3.5664 

t    =-2M2™  log  3.8987 

Red  for  2d  diff.  +  05 

G.  m.  t.,  Oct.  27,      15  48  05 


CONVERSION   OF  TIME.  75 


CHAPTER   V. 

CONVERSION   OF  THE   SEVERAL   KINDS  OF  TIME.— 
RELATION   OF  TIME   AND   HOUR-ANGLES. 

CONVERSION    OF    TIME. 

101.    Problem   23.      To   convert   apparent  into    mean 
time,   or  mean  into  apparent  time. 

Solution.    For  the  same  instant,  let 

Tm  represent  the  local  mean  time ; 

Ta,  the  local  apparent  time ;  and 

JSf  the  equation  of  time  with  the  sign  of  its  application  to 

apparent  time. 
Then,  since  the  equation  of  time  is  the  difference  of  mean  and 
apparent  times  (Art.  67), 


^m  =  Ta  +  JEJ, 


\  }  (61) 


Ta=Tm-  E. 

The  reduction,  then,  is  made  by  finding  from  the  Almanac  the 
equation  of  time  for  a  given  apparent  time,  from  page  I  of 
the  month  (Prob.  16),  or  for  a  given  mean  time  from  page  II 
(Prob.  15),  and  applying  it  to  the  given  time  according  to  the 
precept  at  the  head  of  the  column  where  it  is  found. 

102.    The  equation  of  time  on  page  I  is  sometimes  called 
the  mean  time  of  apparent  noon  ;  and  on  page  II  the  apparent 


76  NAVIGATION. 

time  of  mean  ?won.  Eegarding  it,  as  in  (61),  as  the  reduction 
of  apparent  to  mean  time,  it  indicates,  when  additive  and  in- 
creasing, or  subtractive  and  decreasing,  that  mean  time  is 
gaming  on  apparent  time. 

103.  Problem  24.  To  convert  a  mean  into  a  sidereal 
time  interval,  or  a  sidereal  into  a  mean  time  interval. 

Solution.  The  sidereal  year  is  365.25636  mean  solar  days, 
or  366.25636  sidereal  days ;  so  that  the  same  interval  of  time 
which  is  measured  by  365^.25636  reckoned  in  mean  time,  is 
measured  by  366d.25636  if  reckoned  in  sidereal  time.  Since 
both  are  uniform  measures  of  time,  if  we  represent  any  inter- 
val by 

t,  if  expressed  in  mean  time, 
s,  if  expressed  in  sidereal  time,  then 
g  =  366.25636  =  10Q27379 
t       365.25636  ' 

whence 

8  =  1.0027379  t  =  t  +  .0027379  t,  (62) 

t  =  0.9972696  s  =  s  -  .0027304  s,  (63) 

by  which  the  reduction  from  one  to  the  other  may  be  made. 

The  computation  is  facilitated  by  Table  II  of  the  American 
Ephemeris,  for  converting  sidereal  into  mean  solar  time,  which 
contains  for  each  second  of  s  the  value  of  .0027304  s ;  and  by 
Table  III,  for  converting  mean  solar  into  sidereal  time,  which 
contains  for  each  second  of  t  the  value  of  .0027379  t. 

Tables  8  and  9  (Bowd.)  contain  the  same  quantities. 

104.  If  in  (62)  t  =  24* ;  8  =  24*  Sm  56s.5553  ;  or  in  a  mean 
solar  day  sidereal  time  gains  on  mean  time  3m  56s.5553.  In 
lh  of  mean  time  the  gain  is  9S.8565. 

If  in  (63)  8  =  24* ;  t  =  24*  —  Sm  55s.9094  ;  or  in  a  sidereal 


CONVERSION   OF  TIME.  77 

day  mean  time  loses  on  sidereal  time  Sm  55*.9094.     In  lh  of 

sidereal  time  the  loss  is  95.8296. 

If  t  and  s  in  the  last  term  are  expressed  in  hours  (62),  and 

(63)  become  .   .   ft_  OKaK  . 

v    J  s  =  t  +  9S.8565  t,   ) 

t  =  s-9s.S296s;  \ 

by  which   the    reductions  may   be   more  readily  calculated, 
when  the  tables  are  not  at  hand. 


105.  Problem  25.  To  convert  mean  time  at  a  given 
place  into  sidereal  time. 

Solution.     Let 

A.  represent  the  longitude  of  the  place,  expressed  in  time, 

+  when  west, 
T,  the  local  mean  time, 
S,  the  corresponding  sidereal  time, 
t,  the  interval  from  mean  noon  in  mean  time  (differing 

from  T  only  by  omitting  the  day), 
s,  the  same  interval  in  sidereal  time, 
S0,  the  sidereal  time  of  mean  noon  at  Greenwich, 
S0r,  the  sidereal  time  of  mean  noon  at  the  place ; 
then,   since  X  expresses  the  Greenwich  time  of  local  noon, 

(Art.  92), 

$Q'  =  #0  +  .0027379  X  A 

evidently  #=«  +  #/  I  (65) 

and  by  (62)  s  =    t+  .0027379  t;j 

whence  we  have 

S==    t  +  #0  +  -0027379  (A  +  t).  (66) 

The  Almanac  (page  II)  contains  S0  for  each  Greenwich 
mean  noon,  under  the  head  "  Sidereal  Time."  It  should  be 
taken   out   for   the   given   astronomical   day   of    the   place; 


78  NAVIGATION. 

.0027379  A.  is  then  the  reduction  for  longitude,  additive  in 
west  longitude,  subtractive  in  east.  It,  as  well  as  .0027379  t, 
the  reduction  to  a  sidereal  interval,  may  be  taken  from  Table 
III  of  the  Almanac,  or  from  Table  9  (Bowd.)  ;  or  either  may 
be  computed  by  (62),  or  first  of  (64). 

From  (66),  then,  we  have  the  following  rule : 
To  the  local  mean  time  add  the  sidereal  time  of  Greenwich 
mean  noon  of  the  given  astronomical  day,  the  reduction  of  this 
sidereal  time  for  the  longitude  of  the  place,  and  the  reduction 
of  the  hours,  minutes,  etc.,  of  the  mean  time  to  a  sidereal 
interval. 

The  astronomical  (solar)  day  is  usually  retained.  But  if 
it  be  desirable  to  state  the  sidereal  day,  as  well  as  the  hours, 
etc.,  of  the  sidereal  time,  we  prefix  to  S0  the  sidereal  day  at 
the  instant  of  mean  noon,  which  is  the  same  as  the  astronom- 
ical day  after  the  vernal  equinox  of  each  year  ;  one  day  less 
before  that  date.  At  the  instant  of  the  vernal  equinox  the 
sidereal  time  and  mean  solar  time  coincide.  Before  that  time 
the  mean  sun  transits  before  the  vernal  equinox ;  after  that 
time  it  transits  after  the  vernal  equinox. 

106.  T  +  X  is  the  Greenwich  mean  time.  When  this  is 
given,  or  found  in  the  course  of  computation,  it  will  be  more 
convenient  to  take  out  /S0  for  the  Greenwich  day,  and  the 
combined  reduction,  .0027379  (t  +  X),  for  the  hours,  minutes, 
etc.,  of  Greenwich  mean  time,  instead  of  for  t  and  X  separately. 

It  should  be  noted,  however,  that  in  the  first  method  (Art. 
105),  S0  is  taken  out  for  the  local  day  ;  in  this,  it  is  taken 
out  for  the  Greenwich  day,  provided  X  +  t,  as  used,  expresses 
properly  the  Greenwich  time. 

107.  S0  +  .0027379  (t  -f  X)  is  the  "  sidereal  time  "  of  the 
Almanac  interpolated  for  the  Greenwich  mean  time.     It  is 


CONVERSION   OF  TIME.  79 

more  convenient  to  term  it  the  right  ascension  of  the  mean  sun 
(Art.  93)  ;  and  then  the  translation  of  (66)  will  be,  the  sidereal 
time  is  equal  to  the  right  ascension  of  the  mean  sun  -|-  the  mean 
time. 

This  is  also  evident  from  Fig.  21,  in  which 
P  is  the  pole  ; 
P  M,  the  meridian  ; 
V,  the  vernal  equinox  ; 
T  M,  the  equator. 
T   M  is   also   the   right  ascension 

of  the  meridian,  and  measures 
MP   T,  the   hour-angle   of    T,  or 

the  sidereal  time  (Art.  65). 

If  P  S  is  the  declination-circle  passing  through  the  mean 
sun,  T  Sis  the  right  ascension  of  the  mean  sun,  and 
M  P  S  is  its  hour-angle  or  the  mean  time  (Art.  72),  and  is 

measured  by  the  arc  of  the  equator,  S  M. 
Evidently  TM=tS  +  SM.  (67) 

The  hour-angles  MP  f ,  MPS,  are  reckoned  from  the 
meridian  toward  the  west ;  hour-angles  east  from  the  meridian 
are  then  regarded  as  negative. 

If  P  S  is  the  declination-circle  of  the  true  sun,  then  will 
<f  S  be  the  right  ascension,  and 
MPS  the  hour-angle  of  the  true  sun ;  and 
S  M  will  measure  the  apparent  time, 
and  the  interpretation  of  (67)  will  be,  the  sidereal  time  is  equal 
to  the  right  ascension  of  the  true  sun  -\-  the  apparent  time. 

Examples 

1.  Find  the  sidereal  time  of  1898,  Jan.  30,  10*  Wm  26*.6, 
ast.  mean  time  in  long.  150°  13'  10"  (10*  0m  52'.7)  W. 


80  NAVIGATION. 


FIRST  METHOD. 

SECOND  METHOD. 

h     m      8 

h      m      8 

L.  m.t.,  Jan.  30,  10  15  26.6 

L.  m.  t.,  Jan.  30,     10  15  26.6 

S0 ,                          20  38  58.09 

Long.                     +10   0  52.7 

Red.  for  long.,       +    1  38.71 

G.m.  t.,    Jan.  30,    20  16  19.3 

Red.  of  L.  m.t.,     +    1  41.1 

L.m.t.,                      10  15  26.6 

Sid.  time,                6  57  44.5 

So,                              20  38  58.09 

Red.  for  G.  m.  t.,      +    3  19.81 

Sid.  time,                    6  57  44.5 

2.    Find  the  sidereal  time  of  1898,  Jan.  30,  10*  15"  26s.6, 
ast.  mean  time  in  long.  10*  0m  52s. 7  E. 

h      m      8 

L.m.t.,  Jan.  30,  10  15  26.6 

S0 ,  20  38  58.09 

Red.  for  long  -    188.71} 

Red.  of  L.m.t.,  +     141.10 
Sid.  time,  6  54  27.08 


3.    Find  the  sidereal  time  of  1898,  Sept.  25,  21*  16m  15s,  in 
long.  60°  13'  (=  4*  0m  52s)  W. 


L.m.t.,  Sept. 25, 

21  16  15 

L.m.t.,  Sept.  25, 

21  16  15 

s0> 

12  17  18.26 

Long.,                + 

4    CL52 

Red.  for  long., 

+    0  39.57 

G.  m.t.,  Sept.  26, 

1  17  07 

Red.  of  L.  m.  t., 

+    3  29.66 

So> 

12  21  14.82 

Sid.  time, 

9  37  42.49 

Red.  G.  m.  t., 

+  0  12.67 

Sid.  time, 

9  37  42.49 

4.    Find  the  sidereal  time  of  1898,  Sept.  25,  3*  16w  15s.0,  in 
long.  8*  16m  25s.3  E. 


L.m.t.,  Sept.  25,    3  16  15 

L.  m.  t.,  Sept.  25, 

S0,                          12  17  18.26 

Long., 

Red.  for  long.,       —    1  21.55 

G.m.t.,  Sept.  24, 

Red.  of  L.  m.  t.,    +    0  32.24 

So  i 

Sid.  time,               15  32  43.95 

Red.  for  G.  m.  t., 

Sid.  time, 


3  16  15 

-  8  16  25.3 
18  59  49.7 

12  13  21.71 
+    3  07.25 
15  32  43.96 

CONVERSION   OF  TIME.  81 

108.  Problem  26.  To  convert  sidereal  time  at  any- 
place into  mean  time. 

1st  Solution.     The  sidereal  time  at  mean  noon  at  the  place 

is  from  (65) 

Si  =  Sq  +  . 0027379  X; 

the  sidereal  interval  from  mean  noon, 

s=  S-  S0'=  S-  S0-  .0027379  X ;  (68) 

and  from  (63)  the  corresponding  mean  time  interval, 

t  =  s-  .0027304  s.  (69) 

The  mean  time  T  is  completed  by  prefixing  to  t  the  astronom- 
ical day. 

From  (68)  and  (69)  we  have  the  following  rule : 
From  the  local  sidereal  time  subtract  the  sidereal  time  of 
Greenwich  mean  moon  of  the  given  astronomical  day  and  the 
reduction  of  this  sidereal  time  for  the  longitude  of  the  place  ; 
and  from  the  sidereal  interval  thus  obtained  subtract  the  reduc- 
tion to  a  mean  time  interval ;  and  to  the  result  prefix  the  given 
astronomical  day. 

The  local  sidereal  time  may  be  increased  by  24A  if  neces- 
sary. The  reduction  for  longitude,  .0027379  A,  may  be  taken 
from  Table  III  of  the  Almanac,  or  from  Table  9  (Bowd.)  ;  nu- 
merically, it  is  subtractive  in  west  longitude,  additive  in  east, 
as  applied  to  the  given  sidereal  time.  The  reduction  of  the 
sidereal  interval,  .0027304  s,  may  be  taken  from  Table  II,  or 
from  Table  8  (Bowd.),  and  is  always  subtractive. 

2d  Solution.     Let 

M0  represent  the  "  mean  time  of  the  preceding  sidereal  0h  "  at 

Greenwich ; 
M0 ' ,  the  "  mean  time  of  the  preceding  sidereal  0h  "  at  the 

place ; 


82  NAVIGATION. 

£,  the  interval  from  0h  in  sidereal  time ; 
t,  the  same  interval  in  mean  time : 

then,  since  A.  will  be  the  sidereal  interval  between  the  Green- 
wich and  local  sidereal  0h  (Art.  92), 

Mi  =  M0-  .0027304  A, 
evidently,  T  =  t  •+-  Mi , 

and  by  (63)  *  =  #  -  .0027304  #; 

whence  we  have 

T  =  £  +  M*  -  .0027304  (A.  +  J8 ).  (70) 

The  Almanac  (page  III)  contains,  J^  for  the  Greenwich 
sidereal  0h  on  each  mean  day.  The  Almanac  date  of  the  pre- 
ceding sidereal  0*  is  generally  the  same  as  the  local  astronom- 
ical date  when  the  sidereal  time  is  less  than  the  "  sidereal 
time  at  mean  noon"  (page  II),  but  ld  less  when  the  sidereal 
time  is  greater  than  that  at  mean  noon.  The  doubtful  case  is 
when  the  mean  time  is  within  4m  of  noon:  the  comparison 
must  then  be  made  with  the  sidereal  time  at  the  nearest  local 
mean  noon. 

The  reduction  of  M0  to  the  local  meridian  is  —  .0027304  A., 
which  may  be  taken  from  Table  II,  or  from  Table  8  (Bowd.). 
It  is  subtr active  in  west  longitude,  additive  in  east. 

The  reduction  of  the  sidereal  interval,  .0027304  S,  may  be 
taken  from  the  same  tables ;  it  is  always  subtractive. 

The  combined  reduction,  .0027304  (X  +  >S),  may  be  taken 
out  for  the  Greenwich  sidereal  time,  (X  +  $)>  instead  of  for 
X  and  S  separately;  but  with  these  precautions,  that  when 
X  +  /S  >  24 h,  M0  may  be  taken  out  for  ld  later  than  stated 
in  the  previous  precept,  and  interpolated  for  the  excess 
of  (A.  -f  S)  over  24^ ;  and  when  (A.  +  /S)  is  negative,  to  retain 
its  negative  character,  or  else  take  out  M0  for  one  day 
earlier. 


CONVERSION   OF  TIME.  83 

3d  Solution.     From  (66)  we  have 

t  =  S  -  [#0  +  .0027379  (t  +  A)],  (71) 

so  that,  when  the  Greenwich  mean  time  (t  -f-  A)  is  sufficient^ 
known,  we  may  find  for  it  the  right  ascension  of  the  mean 
sun  (Art.  107), 

S0  +  .0027379  (t  +  A), 

and  subtract  it  from  the  given  sidereal  time:  or,  the  mean 
time  is  equal  to  the  sidereal  time  —  the  right  ascension  of  the 
mean  sun.  So  also  we  have  from  Art.  107  the  precept :  the 
apparent  time  is  equal  to  the  sidereal  time  —  the  right  ascension 
of  the  true  sun. 

Examples. 

1.  1898,  Jan.  30  (ast.  day),  in  long.  10"  0m  52*.7  W.,  the 
sidereal  time  is  6*  57m  448.5  ;  find  the  mean  time. 

h     m      8  h    m     s 

L.sid.t.,                  6  57  44.5  L.sid.t.,  6  57  44.5 

S0  (Jan.  30),    —  20  38  58.09  M0  (Jan.  30),  3  20  28.98 

Red.  for  X,              —  1  38.71  Red.  for  A,  —  1  38.44 

Sid.  int.,                10  17  07.7  Red.  of  sid.  t.,  -  1  08.44 

Red.  of  sid.  int.,      —  1  41.1  L.  m.  t.,  Jan.  30,  10  15  26.6 
L.  m.  t.,  Jan.  30,  10  15  26.6 

2.  1898,  Jan.  30  (ast.  day),  in  long.  10*  0m  52*.7  E.,  the 
sidereal  time  is  6*  54m  278.08  ;  what  is  the  mean  time  ? 

h     m     s  h    m      s 

L.  sid.  t.  6  54  27.08  L.  sid.  t.  6  54  27.08 

S0  (Jan.  30),    -  20  38  58.09  M0  (Jan.  30),  3  20  28.98 

Red.  for  A ,  +1  38.71  Red.  for  A,  +1  38.44 

Sid.  int.,  10  17  17.7  Red.  for  sid.  t.,         -  1  07.9 

Red.  of  sid.  int.,      -  1  41.1  L.  m.  t.,  Jan.  30,  10  15  26.6 
L.  m.  t.,  Jan.  30,  10  15  26.6 

3.  1898,  Sept.  26,  9\  a.m.,  in  long.  4A0W528  W.,  the  side 
real  time  is  9*  37w  428.49  ;  find  the  mean  time. 


84  NAVIGATION. 


h      m      s  h     m      8 

L.  sid.  t.,                9  37  42.49  L.  sid.  t.,  9  37  42.49 

S0  (Sept.  25),   -  12  17  18.26  MQ  (Sept.  25),  11  40  46.62 

Red.  for  X ,                 —  39.57  Red.  for  A ,  -  39.46 

Sid.  int.,                21  19  44.66  Red.  for  sid.  t.,  -  1  34.65 

Red.  of  sid.  int.,     -  3  29.66  L.  m.  t.,  Sept.  25,  21  16  15 
L.  m.  t.,  Sept.  25,  21  16  15 


4.  1898,  Sept.  25,  3A,  p.  m.,  in  long.  8*  16m  25'.3  E.,  the 
sidereal  time  is  15*  32m  43*.95  ;  find  the  mean  time. 

h      m      s  h      m      8 

L.  sid.  t.,      15  32  43.95  L.  sid.  t.,  15  32  43.95 

S0  (Sept.  25),  —  12  17  18.26  M0  (Sept.  24),  11  44  47.52 

Red.  for  A.,      +1  21.55  Red.  for  A,  +1  21.33 

Sid.  int.,       3  16  47.24  Red.  of  sid.  t.,  -  2  32.80 

Red.  of  sid.  int.,    -  32.24  L.  m.  t.,  Sept.  25,  3  16  15 
L.  m.  t.,  Sept.  25,  3  16  15 

RELATION     OF    HOUR-ANGLES    AND     TIME. 

109.  Problem  27.  To  find  the  mean  time  of  meridian 
transit  of  a  celestial  body,  the  longitude  of  the  place  or 
the  Greenwich  time  being  known. 

Solution.  In  the  case  of  the  sun  the  instant  of  meridian 
transit  is  apparent  noon  of  the  place ;  for  which  we  have  (61) 

Tm  =  E,  the  equation  of  time, 

which  can  be  taken  from  page  I  of  the  Almanac,  and  interpo- 
lated for  the  longitude,  which  in  this  case  is  also  the  Green- 
wich apparent  time ;  or  from  page  II,  and  interpolated  for 
the  Greenwich  mean  time.  When  E  is  subtractive,  the  sub- 
traction from  the  number  of  days  can  be  performed. 

The  apparent  right  ascension  of  any  body  at  the  instant  of 
its  meridian  transit  is  also  the  right  ascension  of  the  merid- 
ian, or  sidereal  time.  (Art.  65.)  It  suffices  therefore  to  find 
the  right  ascension  of  the  body,  and,  regarding  it  as  the  side- 
real time,  reduce  it  to  meantime  by  Problem  26. 


RELATION   OF  HOUR-ANGLES  AND   TIME.  85 

The  American  Ephemeris  contains  the  apparent  right  as- 
censions of  two  hundred  principal  stars  for  the  upper  culmi- 
nations at  Washington ;  the  British  Almanac  contains  the 
positions  for  the  upper  culminations  at  Greenwich.  They  are 
reduced  to  any  other  meridian,  when  necessary,  by  interpolat- 
ing for  the  longitude. 

The  right  ascensions  of  the  moon  are  given  for  each  hour, 
and  of  the  planets  for  each  noon,  of  Greenwich  mean  time, 
and  may  be  found  for  a  given  Greenwich  mean  time  by  Prob- 
lem 15.  If,  however,  the  longitude  of  the  place  is  given,  the 
local  mean  time  of  transit  of  the  moon,  or  a  planet,  may  first 
be  found  from  the  Almanac  to  the  nearest  minute  or  tenth 
(Probs.  19,  20) ;  then  for  this  mean  time  the  right  ascensions 
of  the  moon,  or  of  the  planet  (Prob.  15),  and  of  the  mean  sun 
(Prob.  18),  may  be  computed.  Subtracting  the  right  ascen- 
sion of  the  mean  sun  from  the  right  ascension  of  the  moon  or 
planet,  will  give  the  mean  time  of  transit  (Prob.  26,  3d  Solu- 
tion.) If  it  differ  sensibly  from  that  previously  obtained,  the 
process  may  be  repeated  with  this  new  approximation. 

If  the  time  of  transit  has  been  noted  by  a  clock,  or  chro- 
nometer, regulated  either  to  local  or  Greenwich  time,  it  should 
be  used  in  preference  to  the  approximate  time  of  transit  found 
from  the  Almanac  in  computing  the  right  ascensions. 

The  American  Ephemeris  contains  also  the  right  ascen- 
sions of  the  moon  and  principal  planets  at  their  transits  of 
the  upper  meridian  at  Washington.  They  can  be  reduced  to 
any  other  meridian  by  interpolating  for  the  longitude  from 
Washington. 

This  solution  will  give  the  time  of  the  upper  culmination 
of  a  heavenly  body.  To  find  the  time  of  a  lower  culmination, 
12*  may  be  added  to  the  right  ascension  of  the  body,  if  suffi- 
ciently well  known;  or,  as  is  generally  preferable,  12*  may 


86  NAVIGATION. 

be  added  to  the  longitude  of  the  place.  The  instant  of  a  lower 
culmination  on  any  meridian  will  be  that  of  an  upper  culmi- 
nation on  the  opposite  meridian. 

Examples. 
1.    Find  the  times  of  meridian  passage  of  the  moon  and 
Jupiter  for  1898,  June  7   (civil  day),  in   long.  100°  15'  W. 
(Example  1,  Art.  95,  p.  70.) 


D 

h       m 

h     m 

Approx.  m.  t.,  June  6,  14  48.8 

June  7,     6  58.3 

Long.                            +  6  41.0 

6  41.0 

G.m.  t.,            June  6,  21  29.8 

June  7,  13  39.3  = 

13*.665 

3)'sR.  A.,  June  6,  21A, 

h      m      8                  a                                 a 

h      m      a                 8 

19  50  50.93  +  2.5350        A2  _  .0079 

12  04  08.50    0.281 

A2  +  .028 

•°079  x  15=      .002 

^X  6.8=. 008 

60 

24 

4-  2.533 

29.8 

0.289 
13.665 

h    m    8          ( 50.66 

(3.76 

Red.  for  G.  m.  t,    4-  1  15.49   \  22.80 

+  3.95J    .17 

I   2.03 

I   .02 

R.  A.  at  transit,  19  52  06.42 

12  04  12.45 

S0 ,                         4  59  40.55 

5  03  37.11 

Red.forG.m.t.,       3  31.88 

2  14.59 

6V,                        5  03  12.43 

5  05  51.70 

M.  t.  of  transit,                                 June  7. 

,   6  58  20.75 

June  6,  14  48  53.99 

+  2.75 

Diff.  f .  appr.  t.            +  5.99 

In5s99iCh-ofRA"+-253 
ln5-yyj_Ch.ofS0,-.016 

M.  t.  of  transit, 

June  6,  14  48  54.13 

110.    Problem  28.     To  find  the  hour-angle  of  the  sun 
for  a  given  place  and  time. 


RELATION  OF  HOUR-ANGLES  AND   TIME.  87 

Solution.  The  hour-angle  of  the  sun,  reckoned  from  the 
upper  meridian  toward  the  west,  is  the  apparent  time  reckoned 
astronomically  (Art.  72).  Its  hour-angle  east  of  the  meridian 
is  negative,  and  numerically  equal  to  24* —  the  apparent  time. 

A  given  mean  or  sidereal  time  must  then  be  converted  into 
apparent  time  :  for  this,  the  longtitude,  or  the  Greenwich 
time,  must  be  known  approximately. 

111.  Problem  29.  To  find  the  hour- angle  of  the  moon, 
a  planet,  or  a  fixed  star,  for  a  given  place  and  time. 

Solution.    In  Fig.  21,  as  described  in  Art.  104, 

°f  M  is  the  right  ascension  of  the  meridian,  and  measures 

MPf,  the  sidereal  time. 

Let 

P  S  be  the  declination-circle  of  the 

mean  sun,  then 
T  S  is  the  right  ascension  of  the 

mean  sun,  and 
MPS   is    the   mean   time,    and   is 

measured    by   the    arc   of    the 

equator,  S  M. 

Let 
P  M'  be  the  declination-circle  of  some  other  celestial  body ; 

then 
T  M'  is  its  right  ascension,  and 
M  P  M'  is  its  hour-angle,  and  is  measured  by  the  arc  M'  M. 

From  the  figure, 

M,M=rM-<fM'=rS  +  SM-<Y>M'.      (72) 
If        f  S  is  the  right  ascension  of  the  true  sun, 

S  M  will  measure  the  apparent  time. 

From  (72),  then,  we  have  the  following  rule  : 


88  NAVIGATION. 

To  a  given  apparent  time  add  the  right  ascension  of  the 
true  sun;  or  to  a  given  mean  time  add  the  right  ascension 
of  the  mean  sun,  to  find  the  corresponding  sidereal  time. 
Then  from  the  sidereal  time  subtract  the  body's  right  ascen- 
sion; the  difference  is  the  hour-angle  west  from  the  merid- 
ian. If  it  is  more  than  12* ,  it  may  be  subtracted  from  24A : 
the  hour-angle,  then,  is  —  ,  or  east  of  the  meridian.  It  is 
necessary  to  know  the  longitude,  or  the  Greenwich  time, 
sufficiently  near  to  find  the  right  ascensions  of  the  sun  and 
body. 

112.  Problem  30.  To  find  the  local  time,  given  the 
hour-angle  of  the  sun  and  the  Greenwich  time. 

Solution.  The  hour-angle  reckoned  westward  is  itself  the 
local  apparent  time,  which  may  be  reduced  to  mean  or  sidereal 
time  (Probs.  23,  24),  as  may  be  required.  The  Greenwich 
time,  or  the  longitude  of  the  place,  is  needed  only  for  this 
reduction. 

113.  Problem  31.  To  find  the  local  time,  given  the 
hour-angle  of  some  celestial  body  and  the  Greenwich 
time. 

Solution.  Find  from  the  Almanac  for  the  Greenwich  time 
(Prob.  15),  the  right  ascension  of  the  body.     Then,  from  (72), 

We  haVC  T  M  =  T  M'  +  M'  M, 

from  which,  and  Arts.  105,  107,  we  have  the  following  rule, 
regarding  hour-angles  to  the  east  as  negative : 

To  the  right  ascension  of  the  body  add  its  hour-angle ;  the 
result  is  the  sidereal  time.  Prom  this  subtracting  the  right 
ascension  of  the  true  sun  gives  the  apyparent  time ;  or  the  right 
ascension  of  the  mean  sun  gives  the  mean  time. 


RELATION   OF  HOUR-ANGLES  AND   TIME.  89 

The  Greenwich  time  is  needed  for  finding  the  required 
right  ascensions. 

If  the  longitude  of  the  place  is  given,  but  not  the  Green- 
wich time,  we  may  first  use  an  estimated  Greenwich  time,  and 
then  revise  the  computations  with  a  corrected  value,  until  the 
assumed  and  computed  values  sufficiently  agree. 

114.     Examples. 

1.  1898,  Jan.  12,  12*  15w  17s.6,  mean  time  in  long.  150° 
13'  10"  W.,  find  the  hour-angle  of  the  moon. 

h      m      a  A      to      « 

L.  m.  t.,  Jan.    12,  12  15  17.6  L.  m.  t.,  Jan.  12,    12  15  17.6 

Long.,  +10  00  52.7  S0,  19  28  00.06 

G.  m.t.,  Jan.  12,  22  16  10.3    =  16m.17        Red.  for  G.  m.  t.,      +3  39.5 
5'sR.A. 
(Jan.  12,  22*),  11  36  04.83  +    1.9622 

i  19.62 
11.77  L.  sid.  t.,  7  46  57.16 

20 
.14 

D  's  R.  A.  at  date 1136  36.56 

D  's  hour  angle,  —  3  49  39.4 

2.  1898,  Jan.  12,  22*  16m  108.3,  G.  m.  t.,  the  moon's  hour 
angle  is  —  3*  49w  39s.4 ;  find  the  L.  m.  t. 


h      m      s 

D 

's  hour  angle, 

—  3  49  39.4 

D 

»a  R.  A.,  Jan.  12,  22*, 

11  36  04.83  +   1.9622 
16.17 
(  19.62 

Red.  for  G.  m.  t., 

I     .14 

L.  sid.  t., 

7  46  57.16 

£0,  Jan.  12, 

19  28  00.06 

Red.  for  G.  m.  t., 

3  39.50 

L.  m.  t.,  Jan.  12, 

12  15  17.6 

90  NAVIGATION. 

3.    1898,  Jan.  12  (12»  nearly),  in  long.  150°  13'  10"  W.,  the 
moon's  hour-angle  is  —  3h  49m  398.4  ;  find  the  L.  m.  t. 

h      m      s  h     m              m 

Long.,                  10  00  52.7          J)'s  mer.  pass.,  Jan.  12,  15  53.5+1.84 

3)  's  h.  a.,              —  3  49.7         Red.  for  long.,  +  18.4 

ch.ofR.A.,  -7.5  Jan.  12,  16  11.9 


^-ch.of  So,  +0.6 -356.6 

1st  approx.  L.m.t.,  Jan.  12,  12  15.3 
Long.,  +  10  00.8 

1st.  approx.  G.m.t.,  Jan.  12,  22  16.1 

h     m      s 

3>'sh.  a.,  —3  49  39.4 

D  's  R.  A.,     Jan.  12,  22*,     11  36  04.83  +  1.9622 

16.1 


( 

- 19.62 

Red.  for  G.  m.  t., 

+  31.59  j 

11.77 
■      .20 

ch.  in  +  4M7  +  .136 

L.  sid.  t., 

7  46  57.02 

—  S0 ,             Jan.  12, 

19  28  00.06 

—  Red.  for  G.  m.  t., 

3  39.49 

—  ch.  in  +4M7  — .012 

2dL.  m.  t., 

12  15  17.47 

cor.  for  4s.  17        +  .124 

Long., 

10  00  52.7 

2dG.  m.t., 

22  16  10.17 

Diff.  from  1st  G.  m.  t., 

+  4.17 

3d  L.  m.  t.,     Jan.  12, 

12  15  17.6 

NAUTICAL    ASTRONOMY.  91 


CHAPTER  VI. 

NAUTICAL    ASTRONOMY. 

ALTITUDES.-AZIMUTHS.  — HOUR-ANGLES    AND    TIME. 

115.  Nautical  Astronomy  comprises  those  problems  of 
Spherical  Astronomy  which  are  used  in  determining  geograph- 
ical positions,  or  in  finding  the  corrections  of  the  instruments 
employed.  In  general  they  admit  of  a  much  more  refined 
application  on  shore,  where  more  delicate  and  stable  instru- 
ments can  be  used,  than  is  possible  at  sea,  where  the  insta- 
bility of  the  waves  and  the  uncertainty  of  the  sea-horizon 
present  practical  obstacles,  both  to  precision  in  observations 
and  to  the  accuracy  of  the  results,  which  cannot  be  obviated. 

116.  In  the  problems  which  are  here  discussed,  the  follow- 
ing notation  will  be  employed : 

L  =  the  latitude  of  the  place  of  observation 
h  =  the  true  altitude  of  a  celestial  body ; 
z  =  90°  —  A,  its  zenith  distance ; 
d  =  its  declination  ; 
p  =  its  polar  distance ; 
t  =  its  hour  angle  ; 
Z=  its  azimuth. 

Let  the  diagram  (Tig.  22)  represent  the  projection  of  the 
celestial  sphere  on  the  plane  of  the  horizon  of  a  place : 


92 


NAVIGATION. 


Z,  the  zenith  of  the  place ;  NZS,  its  meridian ; 

P,  the  elevated  pole,  or  that  whose 

name  is  the  same  as  that  of  the 

latitude ; 
M,  the  position  of  a  celestial  body ; 
Z  M  H,  a  vertical  circle ;  and 
PM,a  declination-circle,  through  M. 

Then,  in   the    spherical   triangle 
PMZ, 

P  Z  =  90°  —  X,  the  co-latitude  of  the 
place ; 

P  M  =p  =  90°  —  d,  the  polar  distance  of  M; 
Z  M  =  90°  —  A,  the  complement  of  its  altitude  or  its  zenith 
distance ; 
Z  P  M  =  t,  its  hour-angle ; 
P  Z  M  =  Z,  its  azimuth. 

The  angle  P  M  Z  is  rarely  used,  but  is  sometimes  called 
the  position  angle  of  the  body. 

This  triangle,  from  its  involving  so  many  of  the  quantities 
whioh  enter  into  astronomical  problems,  is  called  the  astro- 
nomical triangle.  As  three  of  its  parts  are  sufficient  to  deter- 
mine the  rest,  if  three  of  the  five  quantities  X,  d,  h,  %  and  Z 
are  known,  the  other  two  may  be  found  by  the  usual  formulas 
of  spherical  trigonometry.  These  admit,  however,  of  modifi- 
cations which  better  adapt  them  for  practical  use.  The  fol- 
lowing articles  point  out  how  X,  d,  A,  and  t  may  be  obtained. 


117.  The  latitudes  and  longitudes  of  places  on  shore  are 
given  upon  charts,  but  more  accurately  in  tables  of  geograph- 
ical positions,  such  as  are  found  in  books  of  sailing  directions, 
and  in  Table  49  (Bowd.).     At  sea  it  is  sometimes  necessary 


NAUTICAL    ASTRONOMY.  93 

to  assume  them  from  the  dead  reckoning  brought  forward 
from  preceding,  or  carried  back  from  subsequent,  determina- 
tions.    (Bowd.,  Art.  155.) 

118.  The  altitude  of  an  object  may  be  directly  measured 
at  sea  above  the  sea-horizon  with  a  quadrant  or  sextant ;  on 
shore,  with  a  sextant  and  artificial  horizon,  or  with  an  altitude 
circle.  All  measurements  with  instruments  require  correction 
for  the  errors  of  the  instrument.  Observed  altitudes  require 
reduction  for  refraction  and  parallax ;  for  semidiameter,  when 
a  limb  of  the  object  is  observed  ;  and  at  sea,  for  the  dip  of  the 
horizon.  The  reductions  for  dip  and  refraction  are  subtractive  ; 
for  parallax,  additive.  Strictly,  the  reductions  should  be  made 
in  the  following  order :  for  instrumental  errors,  dip,  refraction, 
'parallax,  semidiameter.  In  ordinary  nautical  practice  it  is 
unnecessary  to  observe  this  order. 

Following  it  we  should  have, 

1.  The  reading  of  the  instrument  with  which  an  altitude 
is  measured ; 

2.  The  corrected  reading  or  observed  altitude  of  a  limb ; 

3.  The  apparent  altitude  of  the  limb ; 

4.  When  corrected  for  refraction  and  parallax,  the  true 
altitude  of  the  limb ; 

5.  The  true  altitude  of  the  centre. 

Except  with  the  sea-horizon,  the  observed  and  apparent 
altitudes  are  the  same.  For  the  fixed  stars,  and  for  the  plan- 
ets when  their  semidiameters  are  not  taken  into  account,  the 
altitudes  of  the  limb  and  the  centre  are  the  same.  For  the 
moon,  see  Art.  59. 

Unless  otherwise  stated,  the  true  altitude  of  the  centre  is 
the  altitude  which  enters  into  the  following  problems,  and  is 
denoted  by  A. 


94 


NA  VIGA  TION. 


119.  The  hour-angle  of  a  body  can  be  found  when  the 
local  time  and  longitude,  or  the  Greenwich  time,  are  given. 
(Probs.  28,  29.)  For  noting  the  time  of  an  observation,  a 
clock,  chronometer,  or  watch  is  used ;  at  sea,  only  the  last  two ; 
but  it  will  be  necessary  to  know  how  much  it  is  too  fast  or  too 
slow  of  the  particular  time  required. 


120.  The  declination  of  a  body  can  be  found  when  the 
Greenwich  time  is  known.     (Prob.  15.) 

The  polar  distance  of  a  heavenly  body  is  the  arc  of  the 
declination-circle  between  the  body  and  the  elevated  pole  of 
the  place  ;  that  is,  the  north  pole,  when  the  place  is  in  north 
latitude ;  the  south  pole,  when  it  is  in  south  latitude.     If 

P  P'  (Fig.  23)  is  the  projection  of 
the  declination-circle  through 
an  object,  M; 

P,  the  north  pole  ; 

P',  the  south  pole  ; 

E  Q,  the  equator ;  then  the  polar 
distances, 


P  M  =  P  Q-QM  =  90°  -d. 
F  M  =  P'  Q  +  Q  M  =  90°  +  d. 


Fig.  23. 


That  is,  the  polar  distance  is  90°  —  d  or  90°  -f  d,  according 
as  the  pole  from  which  it  is  reckoned  is  N.  or  S.  This,  how- 
ever, is  regarding  declination,  like  the  latitude,  as  positive 
when  N.,  negative  when  S. 

To  avoid,  however,  the  double  sign  in  the  investigation  of 
the  formulas  of  Nautical  Astronomy,  we  shall  in  most  cases 
consider  the  declination,  which  is  of  the  same  name  as  the 
latitude,  as  positive,  and  that  which  is  of  a  different  name 


ALTITUDE  AND  AZIMUTH. 


95 


from  the  latitude,  as  negative;  hence  the  polar  distance  will 

be  represented  by 

*  J  p  =  90°  -  d. 

When  the  declination  is  of  a  different  name  from  the  lati- 
tude, we  have  numerically 

p  =  90°  +  d. 


ALTITUDE    AND    AZIMUTH. 

121.  Problem  32.  To  find  the  altitude  and  azimuth 
of  a  heavenly  body  at  a  given  place  and  time.  (Time- 
Azimuth.) 

Solution,  Find  the  declination  of  the  body  and  its  hour- 
angle  at  the  given  time.     (Probs.  15,  28,  and  29.) 

Then  in  the  spherical  triangle  PMZ  (Fig.  24),  we  have 

N 


given 


to  find 


PZ  =  90°  -  Z, 
P  M  =  90°  -  d, 
ZPM  =  £, 


Z  M  =  90°  -  A, 
P  Z  M  =  Z. 

By  Sph.  Trig.  (122),  (123),  if  in 
the  triangle  ABC  (Fig.  25),  we 
have  given  b,  c,  and  A  to  find  a  and 
B,  we  have 

tan  ft  —  tan  b  cos  A, 

cos  (c  —  ft)  cos  b 
cos  ft 


cos  a  = 


cot  B  = 


sin  (c  —  ft)  cot  A 
sin  ft 


Fig.  25. 


which,  by  substituting  the  corresponding  parts  of  the  triangle 
P  Z  M,  give 


96 


NAVIGATION. 


tan  </>  =  cot  d  cos  t, 

sm^8in^+Z)sillrf, 

COS  <f> 

cot  Z  =  cos  (<ft  +  -Q cot  j 


(73) 


1 


sin  <£ 
If  we  put  <f>  =  90°  —  <fS,  these  become 

tan  <f>r  =  tan  c£  sec  £, 

cos  (d/  —  X)  sin  c? 

sin  A  = ^—. — -f 9 

sin  </>'  I  (74) 

„       sin  (<J>r  —  L)  cot  £ 

cot  Z  = ^ -r^ 

cos  <£ 

which  afford  the  convenient  precept,  <f>  has  the  same  name,  or 
sign,  as  the  declination,  and  is  numerically  in  the  same  quad- 
rant as  t. 

122.    When  t  =  6h,  <f>'  =  90°,  and  the  3d  of  (74)  assumes 
an  indeterminate  form.     But  from  the  1st  we  have 

tan  d 


cot   t  = 


tan  <f>'  sin  t 9 


which,  substituted,  gives 

cot  Z  = 

sin  <p  sin  t 

which  may  be  used  when  t  is  near  6h. 


sin  (<f>  —  X)  tan  d 


(75) 


123.  h  is  the  true  altitude  of  M.  If  the  apparent  altitude 
is  required,  the  parallax  (Art.  54)  must  be  subtracted,  and 
the  refraction  (Art.  41)  added. 

It  is  sometimes  necessary  to  compute  'the  altitude  of  one 
or  both  bodies,  to  use  in  connection  with  an  observed  lunar 
distance.  The  rules  for  this  purpose  in  Art.  313  (Bowd.) 
are  derived  from  the  above  formulas.  The  result  is  evidently 
more  accurate,  the  smaller  the  hour-angle  t,  especially  if  the 


ALTITUDE  AND  AZIMUTH.  97 

altitude  is  near  90°.  In  these  rules  it  is  best  to  find  the  "  si- 
dereal time,"  or  "  right  ascension  of  the  meridian,"  from  the 
mean  local  time,  instead  of  the  apparent  (Art.  105). 

Z  is  the  true  bearing,  or  azimuth,  of  the  body,  reckoned 
from  the  N.  point  of  the  horizon  in  north  latitude,  and  from 
the  S.  point  in  south  latitude.  It  is  generally  most  conve- 
nient to  reckon  it  as  positive  toward  the  east,  which  will  re- 
quire in  the  above  formulas  —  Z  for  Z,  since  t  is  positive  when 
west.  Restricting,  however,  Z  numerically  to  180°,  it  may  be 
marked  E.  or  W.,  like  the  hour-angle. 

124.  In  Fig.  24,  if  M  m  be  drawn  perpendicular  to  the 
meridian,  then 

,    P  m  =  <f>  =  90°  -  4>'9 

Zm=  (4>  +  Z)  -90°  =  Z-<£'; 

or,  <£  is  the  polar  distance  of  m, 

<f/,  its  declination, 
L  —  <f>,  its  zenith  distance,  positive,  or  of  the  same  name  as  the 
latitude,  toward  the  equator.  A  convenient  precept  is  to  mark 
it  N.  or  S.,  according  as  the  zenith  is  J5T.  or  S.  of  the  point  m. 
m  falls  on  the  same  side  of  the  zenith  as  the  equator  when 
Z  >  90° ;  at  the  zenith  when  Z  =  90°  ;  and  on  the  same 
side  as  the  elevated  pole  when  Z  <  90°.  It  falls  between 
P  and  Z  only  when  t  and  Z  are  both  less  than  90°. 

125.  In  the  case  of  a  Ursae  Minoris  (Polaris),  whose  polar 
distance  is  1°  25',  the  more  convenient  formulas  derived  from 
(73)  will  be,  since  p  and  <£  are  small, 

<f>  =p  cos  t, 
(which  gives  <f>  within  0".5) 

sin  h  =  sin  (Z  +  $)52^£, 


98  NAVIGATION. 

tan  7  —  tan  P  si"  t cos  <t> 
cos  (X  -f  <£)     ; 
or  approximately, 

A  =  X  +  <£, 

Z  =  />  sin  t  sec  (X  -f-  <£). 

Z  is  a  maximum,  or  the  star  is  at  its  greatest  elongation, 
when  the  angle  Z  M  P  (Fig.  24),  or  Z  n  P  (Fig.  30),  is  90°. 
We  then  have 

sin  Z  =  sin  p  sec  X, 
or  nearly 

Z  =  p  sec  X. 

126.  Problem  33.  To  find  the  altitude  of  a  heavenly- 
body  at  a  given  place  and  time,  when  its  azimuth  is  not 
required. 

Solution.  The  1st  and  2d  of  (73)  or  (74)  may  be  used; 
or,  by  Sph.  Trig.  (4), 

cos  a  =  cos  b  cos  c  -f-  sin  b  sin  c  cos  A, 
we  have         sin  h  —  sin  X  sin  d  -\-  cos  X  cos  d  cos  t.  (76) 

which,  since  cos  t  =  1  —  2  sin2  J  t, 

reduces  to 

sin  h  =  cos  (JO  —  d)  —  2  cos  X  cos  e?  sin2  \  t.        (77) 

(L  —  d)  becomes  numerically  (X  +  <#)  when  X  and  d  are 
of  different  names. 

Table  44  contains  for  the  argument  t  in  column  p.m.  the 
log  sin  i  t  in  the  column  of  sines;  which,  doubled,  is  log 
sin2  \  t.  It  is  well  to  note  this ;  for  mistakes  are  often  made 
by  regarding  the  logarithms  in  this  table  as  log  sin,  log  cos, 
etc.,  of  t  instead  of  J  t. 

127.  Problem  34.  To  find  the  azimuth  of  a  heavenly 
body  from  its  observed  altitude  at  a  given  place.  (Alti- 
tude-Azimuth.) 


ALTITUDE  AND  AZIMUTH. 


99 


Solution.     In  this  the  Greenwich  time  of  the  observation 
must  be  known  sufficiently  near  for  finding  the  declination 

of  the  body.  The  observed  altitude 
must  be  reduced  to  the  true  alti- 
tude. Then  in  the  triangle  P  Z  M 
we  have  given  the  three  sides  to 
find  the  angle  P  Z  M. 

In  the  triangle  ABC,  putting 
s  =  \  (a  +  b  +  c),  we  have,  Sph. 
Trig.  (33), 


00Siv  =  J(8ins8in(s-b)). 

*  V   \      sin  a  sin  c      J 


For  the  triangle  P  Z  M, 

B  =  Z,         a  =  90°  —  h,  h  being  the  true  altitude, 
b  =  p,  the  polar  distance, 
c  =  90°  —  X,  the  co-latitude, 
S  =  90°-J(X  +  A-^), 
5_^  =  90°  -*  (Z+  h  +p), 

and  the  formula  becomes 

P)\ 


/cos  %  (X  +  h -f  p)  cos  \(L  -\-h 
cos  X  cos  A 


cos  \  Z==d\ 

or,  if  we  put  s'  =  £  (X  +  A  +  jt>), 

.    ~        //  cos  /  cos  (&' 

cos  4  Z  =i/    =-^- 

V  \      cos  X  cos  A 

which  accords  with  Bowditch's  rule,  Art.  334. 
In  a  similar  way  we  may  find  from  the  formula 


p) 


(78) 


sin 


J  B       /(sin  (s- a)  sin  (s-c)\ 
V  \  sin  a  sin  c  / 


sin  \Z=J[ CQS  £  (coZ  +  *  ±  ji  sil]  £  (CQ  L  +  h  -  d) ^ 
V  \  cos  X  cos  A 


100  NAVIGATION. 

in  which  coi=  90°  —  L ; 

or,  if  we  put  s"  =  £  (co  L  +  A  -f  J), 

SiniZ  =  v/fcoss"si°(s/,-^)\l 
V  \       cos  L  cos  A       / 

(78)  is  preferred  when  Z  >  90°  ;  (79)  when  Z  <  90°. 
If  the  body  is  in  the  visible  horizon,  then  nearly 
h  =  -  (36'  30"  +  the  dip). 

128.  If  the  bearing  of  the  body  is  observed  with  a  com- 
pass at  the  same  time  that  its  altitude  is  measured,  or  if  the 
bearing  is  observed  and  the  local  time  noted,  the  error  of  the 
compass  can  be  found.  For  the  true  azimuth,  or  bearing, 
of  the  body  can  be  found  from  its  altitude  (Prob.  34),  or  from 
the  local  time  (Prob.  32) ;  and  the  compass  error  is  simply 
the  difference  of  the  true  and  compass  bearings  of  the  same 
object,  determined  simultaneously  if  the  object  is  in  motion. 
It  is  marked  JEJ.  when  the  true  bearing  is  to  the  right  of  the 
compass  bearing,  W.  when  the  true  bearing  is  to  the  left  of 
the  compass  bearing.     (Bowd.,  Art.  323.) 

[In  the  triangle  P  M  Z  (Fig.  26),  representing  the  positive 
angle,  P  M  Z,  by  M,  we  have  by  Napier's  "  Analogies  " 

tan  J  (Z  —  m)  =  cot  \  t  sin  \  (L  —  d)  sec  J  (L  -\-  d)  ] 

tan  J  (Z  +  m)  =  cot  J  t  cos  \  (X  —  d)  cosec  J  (L  +  d)  J     ^    ' 

The  Azimuth  Tables  (Hydrographic  Office,  No.  71),  issued  by 
the  Bureau  of  Navigation,  were  computed  by  means  of  (80). 
From  them  the  azimuth  of  any  heavenly  body  whose  declina- 
tion does  not  exceed  23°  may  be  found,  its  hour-angle,  decli- 
nation, and  the  latitude  being  known. 

These  tables  afford  a  very  simple  as  well  as  accurate 
method  of  ascertaining  the  azimuth,  and  are  therefore  spe- 
cially valuable  for  the  usual  compass-work  on  board  ship.] 


ALTITUDE  AND  AZIMUTH. 


101 


129.  The  amplitude  of  a  heavenly  body  when  in  the  true 
horizon  is  its  distance  from  the  east  or  the  west  point,  and 
is  marked  N.  or  S.,  according  as  it  is  north  or  south  of  that 
point ;  it  is,  then,  the  complement  of  the  azimuth. 

Problem  35.  To  find  the  amplitude  of  a  heavenly  body 
when  in  the  horizon  of  a  given  place. 

Solution.  Let  the  body  be  in 
the  horizon  at  M  (Fig.  27),  A  = 
W  M,  its  amplitude.  The  trian- 
gle P  M  N  is  right  angled  at  JT, 
and  there  are  given 

PN  =  X, 

P  M  =  90°  -  d, 

to  find 

N  M  =  Z  =  90°  -  A. 

We  have  cos  P  M  =  cos  P  N  cos  N  M, 

or  sin  d  =  cos  L  cos  Z, 

whence  cos  Z  —  sin  A  =  sin  d  sec  Z,  (81) 

as  in  Bowditch,  Art.  326.  By  (81)  A  is  N.  or  S.  like  the 
declination. 

As  the  equator  intersects  the  horizon  of  any  place  in  the 
east  or  west  points,  it  is  plain  that  the  star  will  rise  and  set 
north  or  south  of  these  points,  according  as  its  declination  is 
N.  or  S. 

Table  39  (Bowd.)  contains  the  amplitude,  A,  for  each  1°  of 
latitude  up  to  70°,  and  each  J°  of  declination  to  30°  computed 
by  (80).  The  convenience  of  this  table,  in  the  case  of  the  sun? 
is  the  only  reason  for  introducing  amplitudes.  It  is  generally 
best  to  express  the  bearing  of  an  object  by  its  azimuth. 


102  NAVIGATION. 

In  this  problem  the  body  is  supposed  to  be  in  the  true 
horizon,  or  about  (36'  +  the  dip)  above  the  visible  horizon. 
Hence  the  rule  to  "  observe  the  bearing  of  the  sun,  when  its 
centre  is  about  one  of  its  diameters  above  the  visible  horizon." 
Or,  the  body  may  be  observed  in  the  visible  horizon  and  a 
correction  (Table  40,  Bowd.)  applied  for  the  vertical  displace- 
ment.    Art.  326  (Bowd.). 

Examples.     (Probs.  32-35.) 

1.  1898,  Jan.  25,  2*  33w  13s  local  mean  time  in  lat.  49° 
30'  S.,  long.  102°  39'  15"  E. ;  required  the  sun's  true  altitude 
and  azimuth.     (74) 

h   m    •      Jan. 25,  O's  dec.      Eq.oft.m    «  s 

L.  m.  t.,  Jan.  25,     2  33  13      -  18°  52' 54".3    +37". 34    -12  37.22-0.561 
Long.  —6  50  37  c  149.4  r2.24 


7.5  +2.40J    .11 

3.4  I    .05 


G-.  m.  t.  Jan.  24,    19  42  36  I     3.4  I   .05 

or  Jan.  25,  —  4.29  18°  55'  34"  —12  34.8 

Eq.  of  t.  —      12  34.8 

L.  ap.  t.  2  20  38.2 

t  =      35  09  33      1.  sec.  0.08748  1.  cot.  0.15221 

d=      18  55  34      1.  tan.  9.53516  1.  sin.      9.51101 

<f>f  =      22  45  14      1.  tan.  9.62264  1.  cosec.  0.41254    1.  sec.  0.03518 
L=      49  30S 

<£'-  L  =  -26  44  46  N  1.  cos.     9.95085    1.  sin.  9.65325  n. 

h=      48  29  30  1.  sin.     9.87440 


Z  =  S  124  43  W  1.  cot.  9.84064  n. 

The  reduction  for  refraction  and  parallax  of  h  =  48°.5  is 
+  45" ;  and  the  apparent  altitude  is  h'=  48°  30'  15".  If  the 
compass  bearing  of  the  sun  at  the  same  instant  had  been 
N.  34°  20'  W.  =  S.  145°  40'  W.,  the  compass  error  would 
have  been  20°  57'  W. 


ALTITUDE  AND  AZIMUTH. 


108 


2.    1898,  July  20,  5*  5Sm  20s  a.m.,  mean  time  in  lat.  38° 
19'  20"  N.,  long.  150°  15'  30"  E. ;  required  the  sun's  azimuth 

(75). 

a     m     s  July  19, 0' s  dec.     Eq.ojt.    „    ,            , 

L.  m.  t.  July  19,  17  58  20  +20°  48'  47"     — 27//.48    -6  02.72  —0.169 

Long.                —10  0102  f 192.4                           ,1.18 

G.  m.  t.  July  19,    7  57  18  J    24.7              -1.34  )    .15 

=  7.955  _3   39'6 1      1.4                          (    .01 

Eq.  oft.  -         6  04.1     +20   45  08.4  -6  04.06 


,.  ap.  t.  July  19,  17  52  15.9 

t  = 

91°  56'  02''  E. 

1.  sec. 

1.47178  n 

1.  cosec. 

0.00025 

d  = 

20  45  08  N, 

1.  tan. 

9.57854 

1.  tan. 

9.57854 

*'=" 

95  05  23  N. 

1.  tan. 

1.05032  n 

1.  cosec. 

0.00171 

L  = 

38  19  20  X. 

<f>'-L  = 

56  46  03  N. 

1.  sin. 

9.92244 

Z  = 

N  72  20  23  E. 

1.  cot. 

9.50294 

Entering  Azimuth  Tables  (p.  89)  with  lat.  38,  dec.  20°  45', 
we  find  by  interpolation  for  1.  ap.  t.  5*  52m  Az.  =  K  72°  14'  E. 
In  the  same  way  with  lat.  39°  (p.  91),  we  find  Az.  =  N.  72°  26' 
E.     .-.  The  true  azimuth  for  38 J°  =  N.  72°  18'  E. 

3.  At  sea,  1898,  May  20,  15*  23™  16s  mean  time  Green- 
wich, in  lat.  40°  15'  S.,  long.  107°  15'  W.,  the  observed  alti- 
tude of  the  sun's  lower  limb  10°  15'  20",  index  correction  of 
sextant  -f-  3'  20",  height  of  eye  18  feet,  bearing  of  sun  by 
compass  1ST.,  41°  45'  E. ;  required  the  sun's  azimuth  and  the 
compass  error  (78). 

G.  m.  t.,  May  20,     15*  23m  16* 
=  May  21,  -8.6 


O 


10  15  20  rl.  c 


+    3  20 

58 -J  S.d.  +  15  50 

I  Par.  +         9 


Dip   —  4  09 
Ref .  -  5  12 


104  NAVIGATION, 


o 


h  =      10  25  18  1.  sec.  0.00723  O's  dec.  +  20  14  55  +  30.21 

L  =     40 15  1.  sec.  0.11734  —4  20  (241.7 

p=    110  10  35  +201035  |   18.1 

2s=    160  50  53 

s  =     80  25  27  1.  cos.  9.22103 

8  —  p  =  -29  45  10  1.  cos.  9.93861 


9.28421 


h  Z=     63°  59'  1.  cos.  \  9.64211 

True  Z=  S  127°  58'  E.  =  N.  52°  02'  E. 
Comp.  bearing  N.  41   45  E. 

Coinp.  error  10   17  E. 

The  1.  ap.  t.  is  8*  18m  nearly.  Entering  Azimuth  Tables 
(p.  178)  with  lat.  40°  and  dec.  20°,  by  interpolation  we  find 
Z  =  S.  127°  58'  E.  In  same  manner  with  lat.  41°  (p.  179) 
we  find,  at  8.18  a.m.,  Z  =  S.  128°  06'  E.  The  true  azimuth  for 
lat.  40J°  S.  is  then,  by  Table,  S.  128°  E.  or  N.  52°  E. 

4.  1898,  Sept.  20,  in  lat.  30°  25'  N.,  long.  50°  16'  W.,  the 
compass  bearing  of  the  0  when  its  centre  was  in  the  visible 
horizon  was  S.  79°  30'  W.  Kequired  the  true  bearing  and 
the  compass  error  (81). 

The  1.  ap.  t.  of  sunset  for  lat.  30  N.  and  dec.  1°  K  is 
6*  02™.     Azimuth  Tables  (Bur.  Nav.),  Table  10  (Bowd.). 


h    m 

L.  ap.  t.,  Sept.  20,     6  02 
Long.                      +  3  21 

O's  dec.  Sept.  20  (Page  I) 
+  0°  59'  06".3  -  58' 

'.35 

G.  ap.  t.,  Sept.  20,     9  23 

=r9.38 

,525 

-    9   07  .4  ]    17 

+  0    49   59       '     4 

.2 
.5 

.7 

a  =          0  50 

1.  sin.  8.16268 

L  =         30  25 

1.  sec.  0.06431 

True  azimuth         =  N.    89  02  W. 

1.  cos.  8.22699 

Comp.  bearing           N.  100  48  W. 
Comp.  error                      11  46  E. 

HOUR-ANGLE  AND  LOCAL   TIME.  105 

Entering  Azimuth  Tables  (p.  72)  with  lat.  30°,  by  interpo- 
lation for  d.  50',  we  get  the  same  result,  Az.  ==  N.  89°  02'  W. , 
with  lat.  31°,  the  Az.  =  N.  89°  03'  W. 

Very  nearly  the  same  result  is  obtained  from  Table  39 
(Bowd.),  by  interpolation. 

HOUR-ANGLE    AND    LOCAL    TIME. 

130.    Problem  36.     To  find  the  hour-angle  of  a  heavenly 
body  in  the  horizon. 

Solution.     In  the  diagram  of  the  last  problem, 

M  P  Z  =  t,  the  hour-angle ; 

and  in  the  triangle  PMN  are  given 

PM=5°-tfJtofind     MPN  =  180°-'' 
We  have  cos  M  P  N  =  tJ 


tan  P  M ' 
whence  cos  t  =  —  tan  d  tan  L.  (82) 

131.  Prom  this  it  is  apparent  that  when  the  latitude  and 
declination  have  the  same  name,  t  >  6A,  and  consequently  that 
2  t,  or  the  time  that  the  body  is  above  the  true  horizon,  >  12A ; 
and  when  the  latitude  and  declination  are  of  different  names, 
t<  6h  and  2  t  <  12*. 

2  t  is  an  interval  of  sidereal  time  for  a  fixed  star,  of  appar- 
ent time  for  the  sun. 

In  the  case  of  the  sun,  t  would  be  the  apparent  time  of  sun- 
set, were  the  refraction  and  dip  nothing,  and  (24ft  —  t )  would 
be  the  apparent  time  of  sunrise. 

Table  10  (Bowd.)  contains  t,  for  different  values  of  L 
and  d. 


106 


NAVIGATION. 


132.  Problem  37.  To  find  the  hour-angle  of  a  heavenly- 
body  at  a  given  place,  and  thence  the  local  time,  when  the 
altitude  of  the  body  and  the  Txreenwich  time  are  known. 

Solution.  Find  the  declination 
of  the  body  for  the  Greenwich 
time,  and  reduce  the  observed  al- 
titude to  the  true  altitude.  Then 
in  the  triangle  P  Z  M  (Fig.  28)  are 
given 

P  Z  =  90°  -  Z, 

PM=jt>, 

Z  M  =  90°  -  A, 

to  find 

ZPM  =  *. 

For  the  triangle  ABC  (Fig.  29),  we  have,  (Sph.  Trig.,  31), 

sin  i  A  =     //sin(.-6)sin(.-c)\ 
y  \  Bin  o  Bin  c  / 

in  which,  putting  A  =  t 
a  =  90°  -  A, 
b=p, 
c  =  90°  -  X, 

we  have  s  -  b  =  90°  -  J  (Z  +  jt>  +  A), 

•—  c=  i  (Z+P~  h)> 
and 

sin  J  t  =  y/7 
or,  if  we  put 


/cos  J  (X  +  p  -f  A)  sin  i  (Z  +p  -  h)\ 
cos  L  sin  j».  / ' 


8'  =  i(X+/>+^ 

.     .   .         //cos  /  sin  (s'  —  A)\ 

sin  i  «  =  4  /    —\ L   , 

V  \      cos  L  sm  p      J 

which  is  Bowditch's  rule,  Art.  262. 


(83) 


HOUB-ANGLE  AND  LOCAL   TIME.  107 

From  Table  44  (Bowd.)  we  may  take  t  directly  from  col- 
umn p.  m.,  corresponding  to  the  log  sin  -J-  t. 

t  is  —  when  the  body  is  east  of  the  meridian. 

When  the  object  is  the  sun  west  of  the  meridian,  t  is  the 
apparent  solar  time;  when  the  sun  is  east  of  the  meridian, 
(24A  —  t)  is  numerically  the  apparent  time. 

When  the  object  is  the  moon,  a  planet,  or  a  star,  we  have 
(Prob.  31),  denoting  its  R.  A.  by  a, 

the  sidereal  time  =  a  -f  i, 
and  the  mean  time     =  a  —  S0'  -f-  t, 

in  which  SJ  is  the  "right  ascension  of  the  mean  sun." 
(Art.  93.)  Or  the  sidereal  time  may  be  converted  into  mean 
time  by  one  of  the  other  methods  of  Problem  26. 

[The  Sunrise  and  Sunset  Tables  (Hyd.  Office,  No.  Ill) 
were  computed  by  applying  the  equation  of  time  to  the  local 
apparent  times  found  by  (83),  assuming  h  =  —  56'  08",  (ref. 
-  36'  29" ;  S.D.  -  16' ;  parallax,  +  9" ;  dip  -  3'  48")  for  lati- 
tudes between  60°  K  and  S. 

From  them  the  local  mean  time  of  sunrise  and  sunset  may 
be  found,  the  declination  and  latitude  being  known.] 

133.    By  the  formula 

//sin  5  sin  (s  —  a)\       a        m  ,    . 

COS  I  A  =  1  /[  r—r\ L  ),        SPH.  TRIG.  (32), 

2  V  y      smb  sin  c      J  v    J 

we  may  obtain  for  the  triangle  P  Z  M  (z  being  the  zenith 
distance), 

\[  cosXsinjt?  J 

or,  putting  s  =  \  (co  L  +  p  +  z) 

(84) 


cos^  =  v/fsin*sin(s-z)\ 
Y  \    cos  L  sin  p    J 


108  NAVIGATION. 

(84)  is  preferable  to  (83)  when  t  considerably  exceeds  6h, 
which  may  be  the  case  in  high  latitudes. 

If  L  =  90 °,  the  horizon  and  equator  coincide,  and  p  -\-  h 
—  90°  and  p  =  z ;  so  that  both  (83)  and  (84)  become  indeter- 
minate. In  very  high  latitudes,  then,  these  equations  approach 
the  indeterminate  form,  and  it  is  impracticable  to  find  with 
precision  the  local  time  from  an  observed  altitude. 

So  also  if  d  =  90°,  the  star  is  at  the  pole  and  L  =  h ;  and 
the  problem  is  indeterminate.  A  great  declination  is  there- 
fore unfavorable. 

134.  If  the  object  is  in  the  visible  horizon  (rising  or  set- 
ting), h  =  —  (36'  -f  dip)  nearly.  With  the  sun,  the  instants 
when  its  upper  and  lower  limbs  are  in  the  horizon  may  be 
noted,  and  the  mean  of  the  two  times  taken  as  the  time  of 
rising  or  setting  of  its  centre.  The  irregularities  of  refraction 
would  affect  nearly  alike  the  dip  and  the  apparent  position  of 
the  sun. 

135.  If  the  time  at  which  the  altitude  is  observed  is  noted 
by  a  watch,  clock,  or  chronometer,  we  may  readily  find  how 
much  the  watch  or  chronometer  is  fast  or  slow  of  the  local 
time.     (Prob.  45.)     For,  let 

G  be  the  time  noted, 

T,  the  local  time  deduced  from  the  observation  : 
c  =  T —  C  will  be  the  correction  of  the  watch  or  chronometer 
to  reduce  it  to  apparent  time,  when  T  is  the  local  appar- 
ent time  ;  to  mean  time,  when  T  is  the  local  mean  time  ;  or 
to  sidereal  time,  when  T  is  the  local  sidereal  time. 

136.  The  observed  altitude  is  affected  by  errors  of  obser- 
vation, errors  of  the  instrument,  and  errors  arising  from  the 
circumstances  in  which  the  observation  is  made ;  such  as  irreg- 


HOUR-ANGLE  AND  LOCAL    TIME. 


109 


ularities  of  refraction  affecting  both  the  position  of  the  body 
and  the  dip  of  the  horizon.  Errors  of  the  first  class  are  dimin- 
ished by  taking  a  number  of  observations.  Thus  several  alti- 
tudes may  be  observed,  and  the  time  of  each  noted  ;  and  the 
mean  of  the  altitudes  taken  as  corresponding  to  the  mean  of 
the  times,  so  far  as  the  rate  at  which  the  body  is  rising  or 
falling  can  be  regarded  as  uniform  during  the  period  of  ob- 
servation.    This  period  should  then  be  brief. 


137.  We  may  easily  find  how  much  a  supposed  error  of  1' 
in  the  altitude  will  affect  the  resulting  hour-angle,  by  dividing 
the  difference  of  two  of  the  noted  times  by  the  difference  in 
minutes  of  the  two  corresponding  altitudes. 

The  effect  will  evidently  be  least  when  the  body  is  rising 
or  falling  most  rapidly.  This  will  be  the  case  when  its 
diurnal  circle  makes  the  smallest  angle  with  the  vertical  cir- 
cle. An  inspection  of  the  diagram  (Fig.  30)  shows  that  this 
is  the  case  when  the  object  is  nearest  the  prime  vertical,  or 
bears  most  nearly  east  or  west. 

Thus  Zn  being  tangent 
to  the  diurnal  circle  nnf9 
the  angle  which  it  makes 
with  it  is  0  ;  and  is  there- 
fore less  than  the  angle 
which  any  other  vertical 
circle,  as  Znf,  makes  with 
nnr. 

The  diurnal  circle  m  mr 
makes  a  smaller  angle  with 
Z  m,  the  prime  vertical,  than 
with  any  other  vertical  circle, 
as  Z  mr. 


110  NAVIGATION. 

The  diurnal  circle  oo'  makes  a  smaller  angle  with  Zo 
than  with  Z  d . 

The  diurnal  circles  make  right  angles  with  the  meridian  ; 
so  that  at  the  instant  of  transit,  the  change  of  altitude  is  0. 

138.  At  sea,  and  to  a  less  extent  on  the  land,  the  latitude 
is  uncertain.  To  ascertain  the  effect  of  an  error  of  V  in  the 
assumed  latitude,  the  hour-angles  may  be  found  for  two  lati- 
tudes separately,  differing,  say,  10' ;  and  the  difference  of 
these  hour-angles  divided  by  10. 

This  is  an  essential  feature  of  Sumner's  method,  which 
will  be  explained  hereafter.  This  method  will  also  show  that 
an  error  in  latitude  least  affects  the  deduced  hour-angle  when 
the  body  is  nearest  the  prime  vertical. 

Examples.     (Prob.  37.) 

1.  At  sea,  1898,  March  20,  10*  15Hi  20s  G.  mean  time,  in 
lat.  41°  15'  S.,  long.  86°  45'  W.  (by  account)  ;  observed  p.m. 
altitude  of  the  sun's  lower  limb  18°  20' ;  index,  cor.  of  sextant 
—  8'  20",  height  of  eye  18  feet ;  required  the  local  mean 
time  (82). 


G.  m.  t.,  Mar  20,  10*  15™ 

20s                O's  dec. 

Eq.  of  t. 

=  10.256 

O         /          //                     // 

-  0  02  04.5  +  59.28 
C  592.8 

+  10  08.1  |    Usl 

m     s                     s 

7  31.56  —  0.749 
(7.49 
-7.68    ]    .15 
'    .04 

4-  0  08  04 

7  23.9 

Q  =    18°  2(Y  00" 
4-50 

r  S.  d.  +  16'  05" 
<  Par.          +  08 

I.  c. 
Dip. 
Kef. 

-  8'  20" 
-4  09 

—  2  54 

• 

4-  16    13 

-15  23 

HOUR-ANGLE  AND  LOCAL   TIME.  Ill 

h  =    18°  20'  50" 

L  =    41  15  1.  sec   0.12387 

p  =    90  08  04  1.  cosec  0.00000 

2  s  =  149  43  54 

s  =  74  51  57  1.  cos  9.41677 

S-h=    56  31  07  1.  sin   9.92120 

9.46184 


L.  ap.  t.,  Mar.  20,    4h  20™  28s.3  1.  sin  £  9.73092 

Eq.  of  t.  +7     23  .9 

L.  m.  t.,  Mar.  20,     4    27     52.2 

Subtracting  the  local  mean  time  from  the  G.  mean  time 
gives  the  long.  +  5A  47"  27s.8  =  86°  52'  W.  If  we  take 
L  =  41°  25'  S.,  we  shall  find  the  1.  ap.  t.  4*  20"1 12-.3 ;  so  that 
for  A  L  =  10'  S,  A  t  =  -  16s. 

At  sea,  1898,  Sept.  2,  Sh  4m  16s,  G.  mean  time,  in  lat.  46° 
16'  K,  long.  152°  0'  E.,  the  observed  altitude  of  the  moon's 
upper  limb,  W.  of  the  meridian,  was  21°  19';  index  cor.  of 
octant,  —  3' ;  height  of  eye,  20  feet ;  required  the  local  mean 
time. 

ft     m      s  o      /      // 

G.  m.  t.,  Sept.  2,  8  04  16         J  21  19  00  D  's  dec. 

=  8  04.27         I.  c.  -    3         +8°  47'  01".8  +13".737 


h    m      s 


r54  .9 

Dip.         -    4  23  +  58  .6  \    2  .7 

I    1  .0 

S.  D.  -  15  55      +8  48  00 

ft'  =    20  55  42      S.  diam.   15'  49"  4-  6' 
Par.  and  Ref .  =     +  51  37     H.  P.        57  55 

h  =    21  47  19 

L  =    46  16     1.  sec    0.16033 


D  's  R.  A.  0  30  09.5  +  2.0957  p  =    81  12     1.  cosec   0.00514 

r8.38 
+  8.9   |    .42        2  a  =  149  15  19 

'    .15  8  =    74  37  40       1.  cos  9.42339 

0  30  18.4  §  -  ft  ^    52  50  20      1.  sin  9.90142 

9.49028 


112  NAVIGATION. 


D  's  H.  A.              4*  30'"  17s     . 

1.  sin  ft 

9.74514 

L.  sid.  t.                 5   00    35.4 

-  S0                     10  46    37.5 

—  Red.  G.  m.  t.            1     19.6 

L.  m.  t.,  Sept.  2,  18   12    38.3 

Long.                — 10  08    22.3  = 

152°  05'  35"  E. 

139.  Problem  38.  To  find  the  hour- angle  of  a  heavenly 
body  when  nearest  to,  or  on,  the  prime  vertical  of  a  given 
place. 

Solution.  If  d  >  Z,  and  with  the  same  name,  as  for  the 
body  whose  diurnal  path  is  n  n'  (Fig.  30),  PZw  will  be 
greatest,  or  nearest  to  90°,  when  Z  n  is  tangent  to  n  n',  and 
consequently  Z  up  =  90°.     We  then  have 

tan  p      tan  L  .    > 

cos  t  = £-  = .  (85) 

cot  L      tan  d 

If  d  <  X,  and  with  the  same  name,  as  for  the  body  whose 
diurnal  path  is  mm',  the  body  will  be  on  the  prime  vertical 
at  m,  and  P  Z  m  =  90°  ;  whence  we  have 

tan  d  /0_x 

cos  t  = .  (86) 

tan  L 

If  d  and  L  are  of  different  names,  the  diurnal  circle  inter- 
sects the  prime  vertical  below  the  horizon,  if  at  all,  and  the 
visible  point  nearest  the  prime  vertical  is  in  the  horizon. 
The  hour-angle  of  this  point  can  be  found  by  (82),  omitting 
the  effect  of  refraction, 

cos  t  =  —  tan  d  tan  L. 

Altitudes  less  than  8°,  however,  are  to  be  avoided. 

If  d  =  L,  the  diurnal  circle  passes  through  the  zenith,  and 
the  body  would  be  on  the  meridian  and  prime  vertical  at 
the  same  instant ;  so  that?  when  d  and  1,  are  nearly  equal, 


HOUR-ANGLE  AJSfl)  LOCAL   TIME.  113 

latitudes  observed  within  a  few  minutes  of  the  meridian 
passage  of  the  body  may  be  used  for  finding  the  time.  It 
is  only  necessary  that  the  change  of  altitude  shall  be  suffi- 
ciently rapid. 

But  when  the  body  is  very  near  the  meridian  in  azimuth 
the  change  of  altitude  is  proportional,  not  to  the  intervals 
of  time,  but  to  the  squares  of  the  hour-angles.  (Art.  150.) 
Hence,  when  the  body  is  in  such  a  position,  the  mean  of 
several  times  does  not  correspond  to  the  mean  of  the  alti- 
tudes. 

From  the  hour-angle  the  local  time  may  be  found  by 
Problems  30  and  31. 

When  the  declination  of  the  body  does  not  exceed  23°,  t 
may  be  taken  from  Azimuth  Tables  (Bur.  Nat.)  with  suffi- 
cient accuracy  for  ordinary  purposes.     (See  Art.  128.) 

Example.     (Prob.  38.) 

1898,  June  25,  in  lat.  40°  15'  N.,  long.  65°  17'  W.,  re- 
quired the  times  when  a  Lyras  and  a  Aquilse  are  on  the 
prime  vertical. 

a  Lyrce.  a  Aquilce. 

L  =  +40°  15'        1.  cot  0.07234  L  =  +  40°  15'       1.  cot  0.07234 

d=+3841.3    1.  tan  9.90354  d=+    836         1.  tan  9.17965 

t  =  q=  lh  15™.  7    1.  cos  9.97588  t  =  ^    5h  18m.8  1.  cos  9.25199 

R.A.  =      18  33  .5  R.A.  =       19   45  .9 

L.  sid.t.  =      17   17  .8  or  19*  49m.2  14   27  .1  or  lh  04™.7 

—  S0'           6    15  .3         6   15  .3  6   15   .3       6   15  .3 

Sid.  int.          11    02  .5       13   33  .9  8    11   .8     18   49  .4 

red.            -1.8          -2.2  - 1    .3         -3.1 
L.  m.  t.,  June  25, 

11   00  .7       13   31   .7  June  25,    8   10.5       18   46  .3 


114 


NAVIGATION. 


CHAPTER  VII. 


LATITUDE. 


140.    Problem  39.     To  find  the  latitude  from   an  ob- 
served altitude  of  a  heavenly  body  on  the  meridian. 

Solution.     Let  the  diagram  (Fig.  31)  be  a  projection  of 
the  sphere  on  the  plane  of  the  meridian  NZ  S: 


Z,  the  zenith; 

N  S,  the  horizon  ; 

P,  the  elevated  pole ; 

PF,  the  axis  of  the  sphere; 

EQ,  the  equator; 

Q  Z,  the  declination  of  the  ze- 
nith, and 

NP,  the  altitude  of  the  pole, 
are  each  equal  to  the  lati- 
tude, L. 


Fig.  31. 


Let 


M  be  the  position  of  the  body; 

QM  =  d,  its  declination  ; 

M  Z  =  z  =  90°  —  h,  its  zenith  distance,  which  it  is  conve- 
nient to  mark  N.  or  S.,  according  as  the  zenith  is  north 
or  south  of  the  body. 

From  the  diagram,  we  have  QZ  =  QM-fMZ 


LATITUDE.  115 

or,  L  =  z  +  d,  (87) 

which  is  the  general  formula. 
If  the  body  is  at  M',  numerically 

L  ==  z  —  d\ 
if  at  M",  Z=d—z; 

or  "  the  latitude  is  equal  to  the  sum  of  the  zenith  distance 
and  declination,  when  they  are  of  the  same  name ;  to  their 
difference,  when  of  different  names ;  and  is  of  the  same  name 
as  the  greater." 

If  the  body  is  at  M"',  or  below  the  pole, 

Q  M'"  =  180°  -  d,       and       L  =  180°  -  d  -  2, 

numerically ;  or  (87)  is  the  correct  formula,  provided  we  use 
180°  —  d,  or  the  supplement  of  the  declination,  instead  of  the 
declination. 

But  in  this  case  we  have  also  from  the  diagram 

L=p  +  h,  (88) 

as  in  Bowditch,  Art.  274. 

The  declination  of  the  body  must  be  found  from  the  Alma- 
nac for  the  time  of  meridian  passage.  (Probs.  17,  21.)  The 
observed  altitude  must  be  corrected  for  dip,  refraction,  etc., 
and  the  true  altitude  derived. 

From  (87)  we  see  that  an  error  of  V  in  the  altitude  will 
produce  an  error  of  V  in  the  resulting  latitude. 

Examples. 

1.  At  sea,  1898,  June  30,  in  lat.  2\°  N.,  long.  105°  18;  W., 
the  observed  meridian  altitude  of  the  sun's  lower  limb  was 
69°  15'  20",  sun  bearing  N. ;  index  cor.  +  3;  20" ;  height  of 
eye,  20  feet ;  required  the  latitude. 


116  NAVIGATION. 


Long. 

-M 

rhlm 

12s  =  7A.02 

0      69° 

15' 

20" 

fin.  cor. 

+  3' 

20' 

+ 

14 

24 

J  S.  diam. 

+  15 

46 

|Dip 

—  4 

23 

I  Ref.  &  p. 

— 

19 

O's  dec. 
23°  1(Y  11".6  N.  -  9". 32 

-  1    5  .2         65".a 

d  =  23   09  06  W. 
29  44  2  =  20  30  16  S. 


L=    2   38  50  N. 


2.  At  sea,  1898,  June  30,  in  lat.  43£°  N.,  long.  150°  15'  E. 
O  =  69°  15'  20'' ;  on  meridian  bearing  S. ;  index  cor.  +  '3'  20" ; 
height  of  eye,  20  feet ;  required  the  latitude. 

Long.  —  10A  lm  0s  =  —  10A.02 

©       69°  15'  20"  /  In.  cor.     +  3'  20"  Q's  dec- 

I  S.  diam.    +15  46  23°  10'  11  ".6  N.  -  9".32 

+  14  24  -,  D.p  _  4  23  +1  33  .2  93".2 

[  Ref.  &  p.  -       19  d  =  23   11  45  N. 

A    =69    29  44  z  =  20   30   16  N. 

£  =  43   42  01  N. 

3.  At  sea,  1898,  Aug.  9,  about  5  a.m.,  in  lat.  17°  40'  S., 
long.  85°  15'  W.,  obs'd  mer.  alt.  of  >>s  upper  limb,  50°  18 ; 
moon  north  ;  index  cor.  —  2' ;  height  of  eye,  16  feet ;  required 
the  latitude. 

Long.  +  5h  41™  =  5A.68 

D  's  mer.  pass.  Aug.  8,  17*  38m.9  +  2™. 03         D  's  S.  diam.  15'  03"+  11" 

f  10"U5 
Red.  for  long.  +11  .5-j     1.22         p'sH.P.     55  08 


.16 

L.  m.  t.,  Aug.  8,             17    50  .4 

G.  m.  t.,  Aug.  8,             23   31   .4 

F  =  50°  18'       (  I.e.  -  2'  00' 

D's  dec.    +  21°  47'  11"9      +  7".165 

-  21.09  )s.d. -15  14 

r  214".  9 

(Dip  -  3  55 

+  03  45       ]       7  .2 

•'       2  .9 

Zi,=490  56'  51" 

d  =  21°  50'  57"  N. 

+  34  40    Par  and  Ref. 

z   =  39   28  29    S.    „ 

h  =50  31  31 

L  =  17   37  32    S. 

LATITUDE.  117 

4.  At  sea,  1898,  Oct.  9,  5  p.m.,  in  lat.  65%°  K,  long.  150° 
E.  ;  obs'd  mer.  alt.  of  a  Lyrae  63°  17',  bearing  S. ;  index  cor. 
-J-  3'  30"  ;  height  of  eye,  17  feet ;  required  the  latitude. 

*'s  alt.  63°  17'  f  In.  cor.  +  3'.5 

_  1  \   Dip         -  4  .0 

h=  63    16  l  **        -     .5 

z  =26  44  N. 
d  =  38  42  N". 
L  =  65   26  N. 

If  the  star  bore  K,  the  latitude  would  be  11°  58'  N. 

5.  At  sea,  1898,  June  18,  in  lat.  23J°  K,  long.  163°  0'  E. ; 
obs'd  mer.  alt.  of  O's  N.  limb  from  N.  point  of  the  horizon, 
89°  50' ;  index  cor.  +  V  20" ;  height  of  eye,  21  feet ;  required 
the  latitude. 


Long.  -  10*  52™  0s  =  -  10*.87  O's  dec.  23°  25'  25"  N".  +  3".02 

0     89°  50'     (  In  cor.     +     1'.3  -  33 

+  12  .6  ]  S.  diam.  +  15  .8 

(Dip  -4.5  a  =  23   24.9  N. 

h  =  90     2 .6  z  =    0     2.6  N. 

L  =  23   27.5  N. 


In  this  example,  the  corrected  altitude  of  the  O's  centre  is 
more  than  90° ;  this  changes  the  sign  of  z. 

6.  At  sea,  1898,  May  18,  in  long.  180°  0'  E.,  the  true  mer. 

alt.   of  the  sun   was  75°  18' ;   sun  bearing  S. ;   required  the 

latitude. 

Long.  -  12*  0/  0"  d  =  19°  30'.  5  N. 

z  =  U  42      N. 
£  =  34   ly.SN. 

7.  At  sea,  1898,  May  17,  in  long.  180°  0'  W. ;  the  true 
mer.  alt.  of  the  sun  was  75°  18' ;  sun  bearing  S. ;  required  the 

latitude. 

Long.  +  12*0*0'  d  =  19°  30'.  5  N". 

z  =  U  42      N. 
£  =  34   12'.5N. 


118  NAVIGATION. 

Examples  6  and  7  are  identical,  the  Greenwich  apparent 
time  being  May  17,  12h  for  both.  They  illustrate  the  neces- 
sity as  well  as  propriety  of  the  rule  for  navigators  near  the 
meridian  of  180°,  to  add  ld  to  the  date  when  they  pass  from 
west  longitude  to  east;  to  subtract  ld  from  the  date  when 
they  pass  from  east  longitude  to  west.  For  instance,  May  18, 
5h  in  long.  180°  15'  E.,  is  identical  with  May  17,  5h  in  long. 
179°  45'  W. 

141.  The  common  mode  at  sea  of  measuring  a  meridian 
altitude  of  the  sun,  is  to  commence  observing  the  altitude  20 
or  30  minutes  before  noon,  repeating  the  operation  until  the 
highest  altitude  is  attained ;  soon  after  which  the  sun,  as  seen 
through  the  sight-tube  of  the  instrument,  begins  to  dip,  or 
descend  below  the  line  of  the  horizon. 

It  is  preferable,  however,  to  find,  from  a.m.  observations 
for  time  and  by  allowing  for  the  run  of  the  ship  in  the  inter- 
val, the  time  of  apparent  noon  by  a  watch,  and  observing  the 
altitude  at  that  time  within  lm  or  2m. 

A  meridian  altitude  of  the  moon,  or  a  star,  can  be  much 
more  conveniently  observed  by  finding  beforehand  the  watch 
time  of  its  culmination,  and  measuring  the  altitude  at  or  very 
near  that  time. 

When  the  sea  is  heavy,  it  is  recommended  to  observe  three 
or  four  altitudes  in  quick  succession,  within  2m  of  the  time  of 
culmination. 

142.  If  the  body  is  changing  its  declination,  or  the  ob- 
server his  latitude,  the  maximum  altitude  is  not  at  the  instant 
of  meridian  passage;  but  after,  if  the  body  and  zenith  are 
approaching;  before,  if  they  are  separating.     Let 

t  be  the  hour-angle  of  this  culminating  point,  in  minutes  ; 


LATITUDE.  119 

A  d,  the  combined  change  *  of  declination  and  latitude  in  lm, 
if  it  is  expressed  in  seconds  ;  or  in  lh  if  it  is  expressed  in 
minutes  / 

A0  h,  the  change  of  altitude  in  lm  from  the  meridian  passage 
due  solely  to  the  diurnal  rotation  (from  Table  26) ; 

A  A,  the  reduction  of  the  maximum  altitude ;  both  expressed 
in  seconds. 
Now  in  the  time  t 

t  A  d  will  be  the  excess  of  altitude  produced  by  the  change  of 
declination  and  latitude ; 

f  A0  A  (as  will  be  shown  in  Art.  150),  the  diminution  of  alti- 
tude due  to  diurnal  rotation ; 

and  we  shall  have 

A  A  =  t  A  d  —  t2  A0  A. 

But  at  a  point  whose  hour-angle  is  2  t,  the  altitude  will  be 
the  same  as  the  meridian  altitude,  or 

0  =  2t&d-  (2*)2A0A; 

Ad 


whence  t  = 


2A0A' 


(89) 
which  accord  with  the  rule  in  Bowditch,  Art.  277. 


and  Ah=  htAd 

2  4  A0A 


Example. 

A  ship  in  lat.  62  K,  on  March  21,  sails  south  14  miles 
per  hour. 

A  d  =  14'  +  r  =  15'  per  hour,  or  15"  per  minute ; 
A0A  =  1".0; 

*  Their  sum,  if  they  both  tend  to  elevate  or  both  to  depress ;  other- 
wise their  difference. 


120  NAVIGATION. 


=  7^;        AA  =  "X  15"  =  56.' 


.-■If -T|P,     A* -Ml 

The  uncertainty  of  altitudes  at  sea  makes  such  a  correc- 
tion of  little  practical  importance;  but  it  is  generally  ne- 
glected by  those  navigators  who  work  out  their  latitudes  to 
seconds,  supposing  that  they  have  attained  that  degree  of 
accuracy.  In  the  above  example,  the  maximum  altitude  of 
the  sun  would  have  been  greater  than  the  meridian  altitude, 
and  the  latitude  obtained  from  it  in  error,  by  nearly  1'.  The 
sun  would  not  have  sensibly  dipped  until  9  or  10  minutes 
after  noon. 

143.  A  difficulty  occurs  at  sea  in  measuring  the  meridian 
altitude  of  the  sun  when  it  passes  near  the  zenith,  on  account 
of  its  very  rapid  change  of  azimuth ;  the  change  being  made 
from  east  to  west,  180°,  in  a  very  few  minutes. 

What  is  wanted  is  the  angular  distance  of  the  sun  from 
the  N".  or  S.  points  of  the  horizon.  One  of  these  points  may 
be  sufficiently  fixed  by  means  of  the  compass,  and  then  the 
angular  distance  from  this  point  observed  within  lm  or  2m  of 
the  meridian  passage  as  determined  by  a  watch  regulated  to 
apparent  time. 

144.  From  (87)  we  have 

z  =  Z  —  d,  (90) 

by  which  the  zenith  distance  may  be  found  when  the  latitude 
and  declination  are  given. 

Also  d  =  L  —  z,  which  may  be  used  at  sea  for  estimating 
the  declination  of  a  bright  star  from  its  estimated  meridian 
altitude.  If  the  time  when  it  is  near  the  meridian  be  also 
noted,  and  converted  into  sidereal  time,  we  have  the  right 
ascension  and  declination  of  the  star  sufficiently  near  for 
determining  what  star  it  is. 


LATITUDE. 


121 


Example. 

July  16,  Sh  45m,  in  lat.  11°  N.,  a  bright  star  is  seen  near 
the  meridian  S.,  at  an  estimated  altitude  of  55°. 


L.  m.  t.  July  16    Sh  45™ 

i  =  ll°N. 

S0                           7  37 

z  =  35  N. 

L.  sid.  t.               16  22 

d  =  24  S. 

The  R.  A.  of  a  Scorpii  (Antares)  is  16*  23m,  and  its  declina- 
tion 26°  13'  S. 

145.  Problem  40.  To  find  the  latitude  from  an  alti- 
tude of  a  heavenly  body  observed  at  any  time,  the  local 
time  of  the  observation  and  the  longitude  of  the  place 
being  given. 

1st  Solution.  Reduce  the  observed  altitude  to  the  true  al- 
titude, and  from  the  local  time  and 
longitude  find  the  declination  and 
hour-angle  of  the  body.  (Probs. 
21,  28,  29.)  Then  in  the  triangle 
P  Z  M  (Fig.  32)  there  are  given 

ZPM  =  «, 
P  M  =  90°  -  d, 
Z  M  =  90°  -  h, 
to  find 

P  Z  =  90°  -  L. 

By  Sph.  Trig.   (146),  if  in  the  triangle  ABC  (Fig.  33) 
are  given  a,  b,  and  A,  we  find  c  by  the  formulas 

tan  <f>  =  tan  b  cos  A, 

,,      cos  d>  cos  a 

cos  <f>  = - — - — , 

cos  b 

which,  applied  to  the  triangle  P  Z  M, 
give 


122  NAVIGATION. 


tan  <j>  =  cot  d  cos  t, 


.,      cos  <t>  sin  h 

cos  <f>  = - —  , 

sin  d 

90°  -  L  mm  <f>  ±  <£'. 


(91) 


These  may  be  changed  into  a  more  convenient  form  for 
practice,  if  we  put  <f>  =  90°  —  <£" ;  then 


tan  <f>"  =  tan  d  sec  £, 

.,       sin  <t>"  sin  A 

cos    <f>  = ■? , 

sin  « 

Z  =  4,"  =f  *'. 


(92) 


Here,  observing  that  -f  and  —  may  be  rendered  by  N.  and 
S.  respectively,  we  mark  <j>"  N.  or  S.  like  the  declination,  and 
<f>  either  N.  or  S. ;  then  the  sum  of  <f>"  and  <f>'  when  of  the 
same  name,  their  difference  when  of  different  names,  is  the 
latitude,  of  the  same  name  as  the  greater.  There  are  two 
values  of  L  corresponding  to  the  same  altitude  and  hour- 
angle,  but  which,  unless  tf  is  very  small,  will  differ  largely 
from  each  other ;  so  that  we  may  take  that  value  which  agrees 
best  with  the  supposed  latitude  (at  sea  the  latitude  by  ac- 
count).    When  t  >  6*,  $>"  >  90°,  as  in  (74). 

<f>  is  positive  if  Z  >  90°,  and  negative  if  Z  <  90° ;  the  sign 
of  <j>  may  therefore  be  determined  by  the  bearing  of  the  body. 

146.  In  Fig.  32,  if  M  m  be  drawn  perpendicular  to  the 
meridian,  we  shall  have 

<f>    =P»i,  the  polar  distance  of  ra, 

4>"  =  90°  -  P  ra,  the  declination  " 

<J>'  =  Z  m,  the  zenith  distance    " 

When  <f>  is  very  small  (that  is,  when  M  m  nearly  coin- 
cides with  M  Z),  <f>'  cannot  be  found  with  precision  from  its 


LATITUDE.  123 

cosine.  If  not  greater  than  12%  it  can  be  found  only  to  the 
nearest  minute  with  5-place  tables  ;  if  only  2°,  it  can  be 
found  only  within  3'.  The  more  nearly,  then,  that  M  m  coin- 
cides with  Z  m,  or,  in  other  words,  the  nearer  the  body  is  to 
the  prime  vertical,  the  less  accurate  is  the  determination  of 
the  latitude.  If  the  body  is  on  the  prime  vertical,  </>'  cannot 
be  found  within  30'. 

147.  To  find  the  effect  of  an  error  in  the  altitude,  differen- 
tiate the  second  equation  of  (91),  regarding  <f>  and  h  as  varia- 
bles ;  and  we  have 

d  n  =  - sin  *"  coahdh 

sin  d  sin  <j> 

or,  since 

cos  <j>  _  sin  <f>f/ 

sin  h        sin  d  ' 

d  f  =  -  d  A.  cot  $  cot  h.  (93) 

But  in  the  triangle  MZm, 

iv/r  rj  M7T)       tan  m  Z 

cos  MZm=-  cos  M  Z  P 


tan  M  Z ' 


that  is,  Z  being  the  azimuth, 


—  cos  Z  = ?-,  or  sec  Z  =  —  cot  d>  cot  h,         (94) 

cot  h  ,<  N    ' 

and  therefore, 

d  <£'  =  d  A  sec  Z.  (95) 

If  the  body  is  on  the  meridian,  Z  =  0  or  180°,  and  numer- 
ically d  <j>'=  d  A. 

The  nearer  Z  is  to  90°,  the  greater  is  d  <J>'.  If  Z  =  90°,  or 
the  body  is  on  the  prime  vertical,  sec  Z  =  oo,  and  d  <//  is  in- 
calculable.    If  Z  is  near  90°,  (95)  is  inaccurate. 


124  NAVIGATION. 

A  star  which  transits  the  meridian  near  the  zenith,  changes 
its  azimuth  very  rapidly.  Unless  observed  on  the  meridian, 
it  cannot  be  depended  on  for  latitude. 

148.  To  find  the  effect  of  an  error  in  the  time,  and 
consequently  in  the  hour-angle,  we  may  take  the  form- 
ula (76):  — 

sin  h  =  sin  L  sin  d  -\-  cos  L  cos  d  cos  t, 
and  differentiate  regarding  L  and  t  as  variables  ;  which  gives, 

,  -r  cos  L  cos  d  sin  t  -,  . 

a  ±i  = : — = =— : — -  .  a  t. 

sin  Jj  cos  a  cos  t  —  cos  L  sin  d, 

But,  cos  d  sin  t  =  cos  h  sin  Zy  \  Sph.  Trig. 

and     sin  Z  cose?  cos  £  —  cos  X  sin  d=  cos  A  cos  Z  ]      (114). 

so  that, 

d  L  =  —  15  d  t  cos  L  tan  Z,  (96) 

which  requires  that  the  azimuth  should  be  known. 

At  sea  the  chief  uncertainty  of  this  problem  is  in  tne  time, 
either  from  its  imperfect  determination  by  observation,  or 
from  unavoidable  errors  in  allowing  for  the  run  of  the  ship 
in  the  interval  between  the  observations  for  time  and  for 
latitude. 

By  (96)  it  appears  that  the  effect  of  an  error  in  the  time 
is  0  when  Z=0  or  180°,  that  is>  when  the  body  is  on  the 
meridian  ;  and  the  effect  is  incalculable  when  Z  =  90°  or  270°, 
or  the  body  is  on  the  prime  vertical. 

Moreover,  the  effect  is  opposite  on  different  sides  of  the 
meridian,  and  would  be  eliminated  by  two  observations  of  the 
same  body,  or  of  different  bodies,  at  the  same  azimuth  E.  and 
W.  of  the  meridian. 


LATITUDE.  125 

149.  2d  Solution.  If  the  latitude  is  already  approxi- 
mately known,   we  have  (76) 

sin  A  =  sin  L  sin  d  -j-  cos  L  cos  d  cos  t ; 
whence 

cos  (X  —  d)  =  sin  A  +  2  cos  L  cos  6?  sin2  £  t ; 

or  since  (L  —  d)  is  the  meridian  zenith  distance  of  the  body, 

(87),  denoting  it  by  z0,  and  the  meridian  altitude  by  A0,  we 

have 

cos  z0  =  sin  A0  =  sin  A  +  2  cos  i  cos  rf  sin2  J  £,         (97) 

in  which  we  may  use  the  approximate  value  of  L  in  comput- 
ing the  term  2  cos  L  cos  d  sin2  £  £,  which  term  is  smaller 
the  nearer  the  observation  is  taken  to  the  meridian.  Having 
found  the  meridian  zenith  distance,  we  may  find  the  latitudes 
as  in  Problem  39.  If  the  computed  value  of  L  differs  largely 
from  the  assumed  value,  the  computation  should  be  repeated, 
using  this  new  value. 

150.  3d  Solution.  Reduction  to  the  Meridian.  When  the 
observation  is  taken  very  near  the  meridian,  we  may  find  the 
correction  to  be  applied  to  the  observed  altitude  to  reduce  it 
to  the  meridian  altitude,  thus : 

From  (97)  we  have 

sin  h0  —  sin  h  =  2  cos  L  cos  d  sin2  \  t, 
whence,  by  Pl.  Trig.  (106), 

cos  J  (A0  +  A)  sin  \  (A0  —  A)  =  cos  L  cos  d  sin2  -J  t. 
But  A0  and  A  differing  very  little,  we  may  put 

cos  i  (A0  -J-  A)  =  cos  A0  =  sin  z0  =  sin  (Z  —  d), 


so  that 


,  ,,         _.       cos  L  cos  d  sin2  4-  t  ,    „ 

3mH*o-A)=       gin(Z_rf)*    ■  («) 


126  NAVIGATION. 

Put  A  h  =  h0  —  h,  the  reduction  of  the  observed  to  the 
meridian  altitude,  or,  as  it  is  usually  called,  The  reduction  to 
the  meridian  ;  and,  since  A  h  and  t  are  quite  small,  put 

sin  J  A  h  —  -J  A  h  sin  1"  (A  h  being  expressed  in  seconds  of  arc), 

sin  \t  =  \t  X  15  sin  1" (t     "  "  "     of  time), 

then  (98)  reduces  to 

.   ,       112.5  sin  1"  cos  L  cos  d 
sin  (L  —  d) 

or,  since  sin  1"  =  0.000004848, 

A  .      0/r.000545  cos  Z  cos  d       ... 

A  A  = : — 7-= — — r;^ X  t*  (t  w  seconds). 

sin  (L  —  d)  v  J 

In  this  formula  t  is  in  seconds  of  time ;  but  if,  as  is  usual, 
t  is  expressed  in  minutes,  we  must  put  (60  t)2  for  t2,  so  that 

we  have 

A  .       1".96349  cos  L  cos  d 

Ah= :— ry ^r X«2  (99) 

sin  (L  —  d)  K    J 

If  ^  =  lm,  the  formula  expresses  the  change  of  altitude  in 

one  minute  from  the  meridian.     Representing  this  by  A0  h,  we 

have 

A    .      1".96349  cos  L  cos  d  ,      \ 

\h=  — Tf T\ (10°) 

0  sin  (L  —  d) 

Ah  =  t2\h, 
and  (101) 

h Q  —  h  +  A  h,  the  meridian  altitude. 


J 


Whence   the   latitude    is    found   as   by   a   meridian   altitude 
(Prob.  39).     Art.  278  (Bowd.). 

Bowditch's  Table  26  contains  the  values  of  A0  h  for  each  1° 
Of  declination  from  0  to  24°,  and  each  1°  of  latitude  from 


LATITUDE.  127 

0  to  70° ;  except  when  L  —  d  <  4°,  for  then  A  A  is  so  large 
that  (99)  and  (100)  become  inaccurate.  In  this  case  the  body 
is  near  the  zenith,  and  altitudes  out  of  the  meridian  do  not 
afford  a  reliable  determination  of  the  latitude. 

Bowditch's  Table  27  contains  t2  for  each  l5  of  t  from  0  to 
13™. 

When  h  is  small,  the  reduction  to  the  meridian  may  be 
found  by  this  method  quite  accurately,  even  when  t  is  as  great 
as  12™.  If  h  is  near  90°,  t  must  be  taken  within  much  nar- 
rower limits.  Indeed,  in  this  case  z0,  or  its  equal  (i  —  d),  is 
very  small,  and  consequently  A0  h  becomes  large.  Unless  then 
t  is  sufficiently  small,  A  h  will  be  too  great  for  the  assumption 
sin  J  A  h  =  A  h  sin  1". 

If  d  >  X,  sin  {L  —  d)  =  sin  z0  is  negative ;  that  is,  z0  will 
have  a  different  name  or  sign  from  L  (Art.  140).  Properly 
h,  h0,  and  A0A  would  also  become  negative  to  correspond. 
Still,  however,  we  shall  have  numerically 

h0  =  h  +  A  h. 

We  may  therefore  disregard  the  sign  of  L  —  d  in  (100), 
and  consider  h  and  h0  as  always  positive. 

If  the  star  is  observed  at  its  lower  culmination,  then  t  will 
be  the  hour-angle  from  the  lower  branch  of  the  meridian,  and 
for  d  we  may  use  180°  —  d  (Art.  140).  A0  h  and  A  h  are  then 
numerically  subtractive. 

Examples.     (Prob.  40.) 

1.  At  sea,  1898,  July  17,  lh  p.m.,  in  lat.  36°  38'  S.,  long. 
105°  18'  E.,  by  account ;  time  by  Chro.,  5"  47TO  14s ;  Q ,  30°  15' ; 
N*.  W'y ;  index  cor.  +  2'  30" ;  height  of  eye,  17  feet ;  Chro. 
cor.  (G.  m.  t.)  -f  14m35;  required  the  latitude. 

(By  128)  - 


128 


NAVIGATION. 


T.byChro.  +  12*,17  47  14 

Chro.  cor.  +  14    3 

G.  m.  t.  July  16    18    117=18.021 

-  Long.  +7    1  12 

L.  m.  t.  July  17      1    2  29 


&sdec.  11tk  Eq.  oft.  VIth 

+  2°1  10  03.6  —25.7  -5  53.51  +6.214 
-  6  -6 

-1.28 


+  2  34.2 

154.2 
+  2112  37.8  -5  54.79 


Eq.  of  t.               -5  54.8  0  30  15         ( I.  c.  +    2  30  Dip               -4  02 

L.  ap.  t.             0  56  34.2  +12  43    Js.  d.  +  15  47  Ref.  &  Par  -1  32 

h  =30  27  43  1.  sin     9.70497 

t*  =  14  08  33  1.  sec  0.01337 

d  =  21  12  38  1.  tan  9.58892        1.  cosec  0.44154 

f '  =  21  48  42  N.  1.  tan  9.60229        1.  sin      9.57002 

f  =  58  37  32  S.  1.  cos     9.71653 
L  =  364850  S. 

If  we  suppose  an  uncertainty  of  3'  in  the  altitude  and  20' 
in  the  longitude,  by  (94),  (95),  and  (96) 

1.  cot  (-h)  0.2305 n  I.  cos  L  9.903 

1.  cot  #        9.7852     — d  t  =  —20'     log        1.301  n 
Z  =  S.  164°  40'  W.    1.  sec  Z       0.0157  n  1.  tan  Z  9.438  n 

d/i^+3'  log  0.477      — di=+4,.4  1og        0.642 

di=-3,.l  log  0.493  n 

That  is,  an  increase  of  3'  in  the  altitude  will  numerically 
decrease  the  latitude  3'.1 ;  and  a  numerical  increase  of  20'  in 
the  assumed  longitude  will  increase  the  latitude  4'.4.  This 
may  be  conveniently  expressed  in  the  following  way : 

Long.       105°  18'     ±  20'  E. ;  Q,  30°  15'  i  & 
L  =    36°  48'.8  i    4'.4  =f  3M  S. 

*  Instead  of  changing  t  into  arc,  we  may  enter  col.  p.m.  of  Table  44 
with  2  t  =  1*  53m  22s. 


LATITUDE. 


129 


2.     At  sea, '1898,  Dec.  6,  about  5  a.m.,  in  lat.  50°  30'  K, 
long.  135°  25'  W.  (by  account),  time  by  Chro.  2h  00™  52s ;  Chro. 


cor.  (G.  m.  t.)  - 
index  cor.  —  3' 
tude.     (92.) 


12™  34s ;  Obs'd  alt.  of  Mars,  58°  10'  S.  W'y  ; 
height   of   eye,  19  feet ;    required  the  lati- 


T.  by  chro. 

Chro.  cor. 

G.  m.t.,Dec.  6, 

S0 

Ked.  for  G.  m.  t. 
G.  sid.  t. 
Long. 
L.  sid.  t. 
Mars'  K.  A. 
t     = 


2  00  52 
-  12  34 
1  48  18  = 

17  01  10.2 

17.8 

18  49  46 
9  0140 
9  48  06 

8  47  17.3 
100  48.7 


1A.8 


h     m       s  s 

Mars'  R.A.        8  47  16.3  +      0.578 

+  1.0  (   .58 

8  47  17.3  )    .46 


t 

a 

4? 

4>' 


=  15°  12'  11" 

=  20  49  38  N. 
=  21  30  53  N. 
=  28  56  35  N. 
=  50   27  28  N. 


1.  sec  0.01548 
1.  tan  9.58025 
1.  tan  9.59573 


Mars'  dec.  +  20  49  29.3  +      4.66 
+  8.4        1 4.7 
+  20  49  37.7         I  3.7 
tf=  58°  10'       (  I.  c.         -3.00 
-   7. 48  J  Dip         —4.16 
(P.  and  R.-. 32 
h  =  58   02.12      1.  sin       9.92860 

1.  cosec  0.44910 
1.  sin  9.56436 
1.  cos     9.94206 


If  d  h  =  +  5'  and  d  A  =  +  15',  d  t  =  -  15';   and  by  (94), 
(95),  and  (96), 


1.  cot  (-h)  =  9.79491  n 

1.  cor  <£'  =  0.25722 

Z  =  N.  152°  29'  W.  1.  sec  Z  =  0.05213  n 
dh  =  +  5'     log.  0.69897 

d^  =  dl=         -  5'.5         0.74110  n 


1.  cos  L  9.80388 

-dt  =  +15'  log  1.17609 
1.  tanZ  9.71679  n 

dl     =  -  5'  log  0.69676  n 


3.  1898,  May  15 ;  in  lat.  41°  30'  K  (approx.) ;  long.  4* 
47™  30s  W.j  obs'd  alt.  O  67°  18',  bearing  S'ly.  j  Chro.  T.  5* 
38™  28* ;  Chro.  cor.  (G.  m.  t.)  -  51-  485.5  ;  i.e.,  -  V  50"  ;  height 
of  eye,  25  ft.     Eequired  the  latitude.     (101.) 


130  NAVIGATION. 

h    m      s 

Chro.  T.    5  38  28  O'.s  dec.  (Page  1L). 

Ch.  cor.    —  51  48.5  +  lb°  56'  20".4     +35". 2 

G.  m.  t.      4  46  39.5,  May  15,  4*.78  r  140".8 

Long.         4  47  30  +2  48  .2]     24  .6 

L.m.t    23  59  09.5,  May  14  +  18   59  08  .6  I       2.8 

Eq.  t.        +3  51  O     67°18/00//     <  S.  D.  +15'  51" 

L.  ap.  t      0  03  00.5,  May  15  +  8  47       ]  Par.  +         4 

t         =          3  00.5  ^=67   26  47       ' 

£2=9     (Table  27.)  t*  A0  h    +32.4 


A0  h  =  3.6  (Table  26.)      h  =  67°  27'  19".4  j  .10 

z  =  22   32  40  .6  1ST.  .02 


ffgX+  3  51.15-0.025 

N".  ~ 

d  =  18   59  08  .6  N.       +3  51 
i  =  41    31  49       N.  I.  c.  -  V  50" 

Dip  —4  54 
Kef.  -      24 

151.  Problem  41.  To  find  the  latitude  from  a  num- 
ber of  altitudes  observed  very  near  the  meridian,  the 
local  times  being  known. 

Solution.  By  (101)  we  see  that  very  near  the  meridian  the 
altitude  of  a  body  varies  very  nearly  as  the  square  of  its  hour- 
angle.  Hence  we  cannot  regard  the  mean  of  several  altitudes 
as  corresponding  to  the  mean  of  the  times,  since  this  is  assum- 
ing that  the  altitude  varies  as  the  hour-angle.     Let, 

hx ,  h2,  A3,  etc.,  be  the  several  altitudes; 
h  >  h  >  h>  etc.,  the   corresponding  hour-angles  expressed 
in  minutes  / 

and  we  have  as  the  reduction  of  each  altitude  to  the  meridian, 
and  the  deduced  meridian  altitude, 

&1h=t'i.A0h  h0  =  h1  +  b1Ji\ 

A2A=«1.A0A  A0  =  A2  + A2M  etc.        (102) 

A3  h  =  t\ .  A0  h  etc.     h0  =  h3  +  A8  h) 

Thus  the  meridian  altitude  may  be  derived  from  each  alti- 


LATITUDE.  131 

tude,  and  the  mean  of  all  these  meridian  altitudes  taken  as 
the  correct  meridian  altitude.  But  the  following  is  a  more 
expeditious  method :  — 

If  n  is  the  number  of  observations,  the  mean  value  of  h0 
will  be 

-   =  At  -f  A2  +  A3  -I hn      AtA  +  A2A  -j-  A3A  j A„A 

0  n  ^ 

or, 

X     ._  ^1  ~T~  "2  H~  '*3  4~    •  •  •  jjjl  _|_  ^1  H~    ^2  H~    ^3  ~j~  •  •  •  tn  £  fo     (10$) 

0  n  n 

Whence  the  rule : 

Take  the  mean  of  the  squares  of  the  hour-angles  in  min- 
utes (Table  27,  Bowd.)  ;  multiply  it  by  the  change  of  altitude 
in  lm  from  the  meridian  (Table  26) ;  and  add  the  product  to 
the  mean  of  the  altitudes.  The  result  is  the  mean  meridian 
altitude  required.  (Bowd.,  Art.  278.)  From  the  meridian 
altitude  thus  found,  deduce  the  latitude  as  from  any  other 
meridian  altitude.  (Prob.  39.)  Strictly,  however,  the  dec- 
lination to  be  used  is  that  which  corresponds  to  the  mean  of 
the  times,  and  the  hour-angles,  t,  are  intervals  of  apparent 
time  for  the  sun,  and  of  sidereal  time  for  a  fixed  star. 

152.  It  is  unnecessary  to  reduce  each  observed  altitude 
separately  to  a  true  altitude ;  as  the  reductions,  excepting 
slight  changes  of  refraction  and  parallax,  are  the  same  for 
all,  and  may  be  computed  for  the  mean  of  the  observed 
altitudes,  and  applied  to  this  mean  with  the  reduction  to 
the  meridian. 

153.  Should  it  be  desirable  to  compare  the  several  obser- 
vations with  each  other,  and  test  their  agreement,  it  will  be 
sufficient  to  compute  the  several  reductions  to  the  meridian, 


132  NAVIGATION. 

Ai  A,  A2  A,  A3  h,  etc.,  and  apply  them  separately  to  the  read- 
ings of  the  instrument;  or  to  the  half-readings  when  the 
altitudes  are  observed  with  an  artificial  horizon  :  applying, 
also,  the  semidiameter  when  both  limbs  of  the  body  are 
observed. 

154.  If  the  altitudes  are  taken  on  both  sides  of  the  me- 
ridian, and  at  nearly  corresponding  intervals,  a  small  error 
in  the  local  time  will  but  slightly  affect  the  result ;  for  such 
error  will  make  the  estimated  hour-angles  and  the  corre- 
sponding reductions  on  one  side  of  the  meridian  too  large, 
and  on  the  other  side  too  small. 

155.  This  method  is  rarely  used  at  sea,  as  a  single  alti- 
tude on  or  near  the  meridian  suffices.  No  increase  of  the 
number  of  observations  will  diminish  at  all  the  error  of  the 
dip,  which  affects  alike  each  observation  and  the  mean  of 
all.*  But  on  land  it  is  preferable  to  measure  a  number 
of  altitudes  at  the  same  culmination  of  the  body,  and  thus 
diminish  the  "error  of  observation."  Altitudes  of  the  sun 
are  used,  but  the  best  determinations  are  from  the  altitudes 
of  a  bright  star.  To  facilitate  the  operations,  and  avoid 
mistaking  one  star  for  another,  it  is  well  to  compute  the 
altitude  approximately  beforehand.     (Art.  144.) 

If  an  artificial  horizon  is  employed,  the  error  of  the  roof 
is  partially  eliminated  by  making  two  sets  of  observations 
with  the  roof  in  reversed  positions. 

156.  If  two  stars  are  observed  which  culminate  at  nearly 
the  same  altitude,  one  north,  the  other  south  of  the  zenith, 

*  Such  an  error  is  called  constant ;  those  which  affect  the  several 
observations  differently  are  called  variable. 


LATITUDE.  133 

the  error  of  the  instrument  is  nearly  eliminated  ;  for  such 
error  (except  accidental  error  of  graduation)  will  make  the 
latitude  from  one  of  the  stars  too  great,  and  that  from  the 
other  too  small  by  very  nearly  the  same  amount;  the  more 
nearly,  the  less  the  difference  of  the  altitudes.  The  error 
peculiar  to  the  observer  is  also  eliminated. 

If  the  observations  are  made  with  an  artificial  horizon, 
the  error  of  the  roof  is  eliminated  if  the  same  end  is  toward 
the  observer  in  both  sets  of  observations. 

157.  Bowditch's  Table  26  extends  only  to  d  =  24°.  If  a 
star  is  used  whose  declination  is  beyond  this  limit,  or  if 
greater  precision  than  the  table  affords  is  required,  A0A 
may  be  computed  for  the  star  and  place  by  (100). 

1".9635  cos  L  cos  d 


A0h  = 


sin  (L  —  d) 


158.  If  the  observations  are  made  at  the  lower. culmina- 
tion of  the  star,  we  have  only  to  use  in  the  formulas  180°  —  d 
instead  of  d.     (Art.  140.) 

The  altitudes  observed  at  the  same  culmination  are  very 
nearly  the  same.  To  render  the  measurements  independent, 
after  each  observation  move  slightly  the  tangent  screw  of  the 
instrument.  With  the  sextant,  it  is  best  to  make  the  final 
motion  of  the  tangent  screw  at  each  observation  always  in 
the  same  direction ;  for  example,  in  advance. 


Example.     (Prob.  41.) 

1898,  May  22,  9A,  circum-meridian  altitudes  of  a  Virginis 
(jSpica)  at  lighthouse  on  St.  George's  Island,  Apalachicola  Bay, 
Florida,  lat.  29°  37'  K,  long.  85°  5'  15"  W. 


134  NAVIGATION. 


T.  BY  CH. 

Sext.  No.  1.    Art.  Hor.No.  1. 

h    m      8                     o       /     // 

3  28  56  2  alt.  99  22  50        A. 

end.       In.  cor.  —  3'  0" 

3124 

26  50 

Bar.  30.04, 

Ther.  73° 

33  36 

30  00 

Chro.  cor.  (L.m.t.)  +  bh  37m  14».55 

34  56 

32  40 

Long. 

+  5  40    21 

37    8 

32  40 

38  58 

35  00        B. 

end. 

42  45 

34  30 

*'sR.  A. 

13*  19™  52*.  23 

44  33 

31  10 

-<S0 

-4   00    32.19 

48  21 

25  50 

—Red.  for  \ 

-  55  .91 

5125 

22  50 

Sid.  int.  from  0h   9  18    24.13 

99  29  26 

Red. 

-1   31.48 

h'=  49  44  43  a  In. . 
-219/    Kef 

2or.  —  1'30"     L.m.t.  of  trans.    9    16  52.65 

—     49    —  Chro.  cor. 

-5   37   14.55 

h  =  49  42  24 

Chro.  t.  of  trans.  3   39  38.1 

Mean 

Sid. 

t 

t               t* 

m       s 

m       8 

-10  42 

-10  44      115.2 

1".9635 

log         0.2930 

8  14 

8  15        68.1 

L  =+29°  37' 

1.  cos      9.9391 

6    2 

6    3        36.6 

d  =-10   38 

1.  cos      9.9925 

4  42 

4  43        22.2 

L-d  =+40    15 

1.  cosec  0.1897 

2  30 

2  30          6.2 

A0  h  =                   2//.59C 

l  log         0.4143 

0  40 

0  40          0.4 

t2  =                 49  .86 

log          1.6979 

+  37 

+  37          9.7 

A  h  =                +2  .10 

log         2.1122 

4  55 

4  56        24.3 

h  =     49°  42'  24" 

8  43 

8  44        76.3 

h0  =     49   44  34 

1147 

11  49       139.6 

z0  =+40    15  26 

49.86 

d  =-10   38  05 
L  =+29   37  21 

159.  Problem  42.  To  find  the  latitude  from  two  alti- 
tudes near  the  meridian  when  the  time  is  not  known. 
Chauvenet' s  Method.* 

The  method  of  reducing  circum-meridian  altitudes  to  the 
meridian,  when  the  time  is  known,  has  already  been  given 
(Prob.  41).     At  sea,  however,  the  local  time  is  frequently  un- 
certain, while  altitudes  near  the  meridian  are  resorted  to  as 
*  Astronomy,  I,  296. 


A0  A  = - — — — ,  the  change  of  altitude  in  lm 


LATITUDE.  135 

next  in  importance  to  meridian  altitudes  for  finding  the  lati- 
tude. 

As  in  Prob.  41,  let  A0  represent  the  meridian  altitude, 

1".96349  cos  L  cos  d 

sin  (L  —  d) 

from  the  meridian  (Table  26,  Bowd.),  and  as  before, 

A  and  A',  the  true  altitudes, 

Tand  T\  the   corresponding  hour-angles  (in  minutes  of 

time), 
t  =  T7'  —  T,  the  difference  of  the  hour-angles, 
T0  =  |  (T7'  +  27),  the  middle  hour-angle. 

By  (99), 

A0  =  A+A0A^,  j 

H-jif  +  MnJ  v    ; 

The  mean  of  these  equations  is 

A0  =  i  (h  +  A')  +  i  (  7" 2  +  T72)  A0  A.  (105) 

But 

/  77'  —  77\2      /  Tr  4-  TV 

i(^2+n  =  ^__A)  +(— y-J  =(*0§  +  2i8> 

which,  substituted  in  (105),  gives 

A0  =  i  (A  +  A')  +  [J  t2  +  ^o2]  A0 A.  (106) 

The  difference  of  the  two  equations  of  (104)  gives 

A  -  A'  =  (I7'2  -  T72)  A0  A  =  2^  A0A. 
Hence, 

Substituting  this  in  (106),  we  have 

K  =  i(h  +  h')  +  (i  ty  a0  h  +  ^r^3'-      (los) 


136  NAVIGATION. 

The  reduction  to  the  meridian,  then,  is  effected  "  by  adding 
to  the  mean  of  the  two  altitudes  two  corrections;  1st,  the 
quantity  (££)2A0A,  which  is  nothing  more  than  the,  common 
reduction  to  the  meridian  (101),  computed  with  the  half- 
elapsed  time  as  the  hour-angle ;  2d,  the  square  of  one-fourth 
the  difference  of  the  altitudes  divided  by  the  first  correction." 
Several  pairs  of  altitudes  can  be  thus  combined,  and  the  mean 
of  the  meridian  altitudes  taken,  from  which  the  latitude  can 
be  obtained  as  from  an  observed  meridian  altitude. 

160.  The  restriction  of  the  method  corresponds  with  that 
of  reduction  to  the  meridian  (Art.  150). *  Quite  accurate  re- 
sults can  be  obtained  with  hour-angles  limited  to  5m  when  the 
altitude  is  80°,  to  25m  when  the  altitude  is  only  10°.  If  the 
interval  t,  however,  exceed  10w,  A0  h  should  be  computed  to 
two  or  three  places  of  decimals,  as  it  is  given  in  Table  26 
(Bowd.)  only  to  the  nearest  0".l. 

The  accuracy  of  the  method  depends  mainly  upon  the  ac- 
curacy of  the  second  correction,  and  therefore  upon  the  pre- 
cision with  which  the  difference  of  altitudes  has  been  obtained. 
The  altitudes,  then,  should  be  observed  with  great  care.  Errors 
of  the  tabulated  dip  and  refraction,  and  a  constant  error  of 
the  instrument,  will  affect  both  altitudes  nearly  alike.  If  the 
altitudes  are  equal,  this  second  correction  becomes  0.  The 
most  favorable  condition  is,  therefore,  that  of  equal  altitudes 
observed  on  each  side  of  the  meridian. 

At  sea,  the  method  is  especially  useful  for  altitudes  of  the 
sun  observed  with  a  clear,  distinct  horizon.  A  long  interval 
between  the  observations  is  to  be  avoided  on  account  of  the 

*  Table  26  (Bowd.)  gives  A0  h  only  to  the  nearest  0".l;  if,  then, 
it  is  taken  from  this  table,  A0  h  t2  may  be  in  error  1",  if  t>4m.  If, 
however,  Ao^  is  computed  to  the  nearest  0".001,  the  error  of  using 
A0  h  t2  will  not  exceed  1",  unless  t  >  20m  and  /i>60°. 


LATITUDE.  137 

uncertainty  of  the  reduction  of  one  of  the  altitudes  for  the 
run  of  the  ship. 

161.    The   hour-angle  of   either  altitude  may  also  be  ob- 
tained approximately  ;  for  we  have  from  (107),  in  minutes, 

T  _  \Jh  -  W) 

°~  HA.* 

and 

(Art.  299,  Bowd.) 


(109) 


Example. 

Sept.  3,  1898,  in  lat.  37°  30'  N.,  long.  5*  W.,  by  account, 
observed  two  altitudes,  near  noon,  for  latitude. 

C.  time  10*  30™  21s;  observed  alt.  0  59°  43'  40"  (South). 

M       u      1Qh  35m  36s.  u  u      M   590  33/  40// 

I.  c.     —  0/30//;  ht.  eye,  18  ft. 

o    /    //  h   m    8            O's  dec. 

h       =  59  43  40               C.  tx  10  30  21             B  s    „ 

h'      =  59  38  40              C.  U  10  35  26  +  7  27  23.8  -  55.15 


4 


1  15  t      =     05  15  —04  35.8  +    5 


=  59  41  10  ~    =     02  37.5    +  7  22  48 


2  2 


1.  c.  -    0  30 

S.  D.  +  15  54 


(U» 


6.89 


Dip  -    4  09  A0h  =     3.06  log  0.48572 

Par.  and  Kef.      -0  30  /A2 


h  =  59  51  55 


=     6.89  log  0.83822 

1st  cor.     =21.08  log  1.32394 


1st  cor.  21.1  [L~aL)    ~  56,25  log  3.75012 

2d  cor.  4  26.8  2d  cor.      =266.80  log  2.42618 


h0  59  56  43 
z0  30  03  17  N. 
d     7  22  48  1ST. 
Lat.  37  26  05  N. 


138  NAVIGATION. 

162.     Problem  43.     To  find  the  latitude  from  an  ob- 
served altitude  of  Polaris  or  the  North  Pole-star. 

Solution.     The  formulas  (91)  of  Prob.  40. 

tan  <j>  =  cot  d  cos  t 

.,       cos  <£>  sin  h 

cos  <f>  = £ 

sin  d 

90°  -  L  =  cf>  -|-  <f>' 

can  be  greatly  simplified  in  the  case  of  the  Pole-star,  since  its 

polar  distance  is  only  1°  25'. 

Putting  d  =  90°  -  p  and  tf  =  90°  -  <£", 

we  have 

tan  <f>  =  tan  p  cos  t 

or  <\>  =  p  cos  t  (within  0".5) 

j  ft  •      z.  cos  <£ 

sin  <£    =  sin  h  - 

cosp 

L  =  <f>"  —  <f>, 

the  2d  value  of  X,  or  (180°  —  <j>"  —  <f>),  being  excluded,  as  it 

exceeds  90°.    p  and  <f>  are  so  small,  that  the  cosine  of  each  is 

nearly  1,  and  consequently 

sin  <£"  =  sin  h       and       <f>"  =  A,  nearly. 
Thus  we  have 

<f>  =p  cos  t 


(110) 


L  =  h-cf>  l 

If  £  is  more  than  6*  or  less  than  18ft,  cos  £  is  negative,  and 
we  have  numerically 

Let  S  represent  the  sidereal  time,  and  a  the  right  ascen- 
sion of  the  star,  then 

t=S—a    and     <f>  =pcos  (S  —  a). 

If  we  consider  the  right  ascension  and  polar  distance  of  the 
star  to  be  constant,  <f>  may  be  computed  and  tabulated  for  dif- 


LATITUDE. 


139 


ferent  hour-angles  of  the  star,  as  in  Table  4,  Appendix,  in  the 
Nautical  Almanac.  Owing  to  the  change  of  right  ascension 
and  declination,  such  a  table  requires  correction  for  each  year. 
It  will  furnish  an  approximate  value  of  the  latitude ;  but  it  is 
more  accurate  to  take  the  apparent  right  ascension  and  decli- 
nation from  the  Almanac,  and  compute  t  and  <f>. 

<f>  may  be  found  approximately  in  the  traverse  table  (Table 
2)  in  the  Lot.  col.,  by  entering  the  table  with  t  as  a  course, 
and  p  as  a  distance. 

Formulas  (111)  may  be  derived 
from  Fig.  34,  by  regarding  P  M  m 
as  a  plane  triangle,  and  Z  m  =  Z  M. 
The  first  produces  no  error  greater 
than  0"5.  The  error  of  the  sec- 
ond is  evidently  greater  the  greater 
the  altitude,  or  the  latitude.  This 
error,  however,  will  not  be  more 
than  0'.5  in  latitudes  less  than  20°, 
nor  more  than  2'  in  latitudes  less 
than  60°. 

163.  We  may  use  (110)  with  more  exactness,  but  these 
formulas  may  be  modified  so  as  to  facilitate  computation. 

Put  <£"  =  h  -f  A  h 

then,  changing  the  2d  of  (110)  to  a  logarithmic  form,  we  have 

log  sin  (A  -\-  A  h)  =  log  sin  h  -f  log  cos  <f>  —  log  cos  p, 


log  sin  (h  +  A  h)  —  log  sin  h  =  log  sec  p  —  log  sec  <£. 

But  A  h  being  very  small,  representing  by  D/y  the  change  of 
log  sin  h  for  V ,  we  have,  with  A  A  in  seconds, 

log  sin  (A  -j-  A  h)  --  log  sin  h  =  A  h  X  Dtl ; 
whence,  by  substituting  in  the  preceding,  we  obtain 


140  NAVIGATION. 

A  h  =  log  secP  —  log  sec  j  =  lQg  cos  j  —  lQg  cosff-      /112) 
ss  a 

The  difference  of  the  log  secants,  or  log  cosines,  of  p  and  <£ 
is  readily  taken  from  the  table  by  inspection.  D/y  for  log  sin 
h  is  usually  given  in  tables  of  7  decimal  places,  and  hence  A  h 
is  readily  found. 

We  have  then        <l>=p  cos  t 


Z  =  h+Ah-<f>  v 


If  Dy  is  the  change  of  log  sin  h  for  1',  then  in  minutes 

T) 


AAJogsec^-logsecf 


164.  Bowditch  *  contains  four  tables  (28  A,  B,  C,  D)  for 
the  reduction  of  altitudes  of  Polaris,  from  which  they  may  be 
found  to  the  nearest  second.     (Art.  287,  Bowd.) 

Altitudes  of  Polaris  may  often  be  observed  at  sea,  with 
some  degree  of  precision,  during  twilight,  when  the  horizon 
is  well  defined,  and  the  latitude  found  from  them  within 
3'  or  4'. 

Example.     (Prob.  43.) 

1.  At  sea,  1898,  March  31,  7*  15m  19*,  mean  time  in  long. 
160°  15'  E. ;  obs'd  alt.  of  Polaris  38°  18' ;  index  cor.  +  3' ; 
height  of  eye,  17  feet :  what  is  the  latitude  ?     (113.) 


L.  m.  t.,  Mar.  31,  1h  15m 

19* 

Long.  —  10A  41™ 

S0                           0   35 

31.3 

h'=  38°  18'  00" 

I.e. 

+  3' 

Red.  for  long.         —  1 

45.3 

-  2'  16" 

Dip 

-  4'  02" 

Red.  of  L.  m.  t.      +1 

11.5 

h  =  38   15    44 

Ref. 

-  1   14 

L.  sid.  t.               7   50 

16.5 

*'sK.A.             1   20 

47.8 

t  =  6   29 

28.7 

t  =  97°  22'  11" 

1.  cos 

9.10813  n 

p  =        73  .9 

log 

1.86864 

i  =  ^-<^  +  38°25,.2 

<£  =       —  9  .5 

log 

0.97677  n 

*  Chauvenet's  Astronomy,  I,  256. 


LATITUDE.  141 

By  Table  IV,  Naut.  Alm.,  App. 

L.  sid.  t.     lh  5CM  h  =       38°  15'.7 

Less  1   21  .8  Cor.  per  Table  IV.      +  9  .9 

H.  a.  6   28  .5  L =  +  38  25 .6 

^  X  3.5  =  1.1     8.8  +  1.1  =  9.9 
0 


142  NAVIGATION. 


CHAPTER   VIII. 

THE   CHRONOMETER.  — LONGITUDE. 

165.  Astronomically  the  longitude  of  a  place  is  the  dif- 
ference of  the  local  and  Greenwich  times  of  the  same  instant. 
It  is  west  or  east,  according  as  the  Greenwich  time  is  greater 
or  less  than  the  local  time.     (Art.  73.) 

The  mean  solar,  the  apparent,  or  the  sidereal  times  of  the 
two  places  may  be  thus  compared. 

166.  A  chronometer  is  simply  a  correct  time-measurer,  but 
the  name  is  technically  applied  to  instruments  adapted  to  use 
on  board  ship.  It  is  here  used  more  generally,  as  including 
clocks  which  are  compensated  for  changes  of  temperature. 

A  mean  time  chronometer  is  one  regulated  to  mean  time ; 
that  is,  so  as  to  gain  or  lose  daily  but  a  few  seconds  on  mean 
time. 

A  sidereal  chronometer  is  one  regulated  to  sidereal  time. 

167.  A  chronometer  is  said  to  be  regulated  to  the  local 
time  of  any  place  when  it  is  known  how  much  it  is  too  fast, 
or  too  slow,  of  that  local  time,  and  how  much  it  gains  or  loses 
daily.  The  first  is  the  error  (on  local  time) ;  the  second  is 
the  daily  rate.  Both  are  -f  if  the  chronometer  is  fast  and 
gaining. 

It  is  preferable,  however,  to  use  the  correction  of  the  chro- 
nometer, which  is  the  quantity  to  be  applied  to  the  chronom- 


THE   CHRONOMETER — LONGITUDE.  143 

eter  time  to  reduce  it  to  the  true  time,  and  its  daily  change. 
Both  are  -f-  when  the  chronometer  is  slow  and  losing. 

They  will  be  designated  by  c  and  A  c. 

A  chronometer  is  said  to  be  regulated  to  Greenwich  time 
when  its  corrections  on  Greenwich  time  and  its  daily  change 
are  known. 

If  c0  is  the  chro.  cor.  to  reduce  to  Greenwich  time,  and  c 
the  chro.  cor.  to  reduce  to  the  time  of  a  place  whose  longitude 
is  A.  (  +  if  west). 

c0=c|A,     or  c  =  c0  —  X ;  (115) 

so  that  the  one  can  readily  be  converted  into  the  other. 

168.  If  the  correction  of  the  chronometer  at  a  given  date, 
and  its  daily  change,  are  known,  the  correction  at  another 
date  can  easily  be  found.     For  let 

c  be  the  given  correction  at  the  date  T, 
c',  the  required  correction  at  the  date  Tr 
t  =  T'  —  T,  expressed  in  days, 
A  c,  the  daily  change ; 

then  <f  =  c  +  t  A  c.  (116) 

t  is  negative  if  the  date  for  which  the  correction  is  re- 
quired is  before  that  for  which  it  is  given. 

If  Ac  is  large,  t  must  include  the  parts  of  a  day  in  the 
elapsed  time. 

A  c  may  be  given  for  two  different  dates,  and  vary  in 
value.  It  may  then  be  interpolated  for  the  middle  date  be- 
tween the  two  of  this  problem. 

Thus,  if  A'c  be  a  second  value  determined  n  days  after 
the  first,  the  daily  variation  of  A  c,  regarded  as  uniform,  will 


144  NAVIGATION. 

be  A  c  —  A  c 


(117) 


n 

Kepresenting  this  by  A2  c,  we  have  for  the  mean  daily  change 

of  the  chronometer  correction  during  the  period  t,  or  that  at 

the  middle  date, 

A  c  -{-  £  t  A2  c, 

and  the  required  chronometer  correction, 

d  =  c  +  t  A  c  +  J  t2  A2  c.  (118) 

When  the  chronometer  is  in  daily  use,  it  is  convenient  to 
form  a  table  of  its  correction  for  each  day  at  a  particular  hour. 
For  a  stationary  chronometer,  the  most  convenient  hour  is  0* 
of  local  time ;  for  a  Greenwich  chronometer,  0h  of  Greenwich 
time.     . 

Examples. 

1.  Chro.  1675,  regulated  to  Greenwich  mean  time ;  1898, 
Jan.  15,  0ft;  correction  +  l*16w25s.O;  daily  change  —  7*.65; 
required  the  correction,  Jan.  26,  6*. 

Jan.  15,  0*,        Chro.  cor.     +  1A16TO25*.0 

-7*.65x  11.25=  -    1   26.1 

Jan.  26,  Qh .  Chro  cor.  +  1  14   58.9 

This  chronometer  is  slow  and  gaining. 

2.  To  find  the  chro.  cor.  to  reduce  to  local  time,  Jan.  26, 
0^,  in  long.  85°16'E. 

Chro.  cor.  (Jan.  26  6*  G.  t. )  +  1*  14™  588.9 

—  Long.  +6  +5  41      4 

Ked.  for  - 12  +  3.8 

Chro.  cor.  (Jan.  26  0  L.  t.)  +  6  56     6.7  or  —  5»3m53*.3 

3.  To  form  a  table  of  chronometer  correction  for  each  day 
from  Jan.  26,  6*,  to  Feb.  6,  6*. 


THE   CHRONOMETER—  LONGITUDE,  145 


G.  M.  T. 

Chbo.  Cob. 

G. 

M.  T. 

Chbo.  Cob. 

Jan.  26  6* 

+  1A14™588.9 

Feb 

.  1  6h 

+  1A  Um  13s.O 

27  6 

14    51.3 

2  6 

14     5.4 

28  6 

14    43.6 

3  6 

13   57.7 

29  6 

14    36.0 

4  6 

13   50.1 

30  6 

14    28.3 

5  6 

13    42.4 

31  6 

+  1  14    20  .7 

6  6 

+  1  13    34  .8 

169.  To  find  the  rate,  or  daily  change,  of  a  chronometer, 
it  is  necessary  to  find  the  correction  of  the  chronometer  on 
two  different  days,  either  from  observations,  or  by  comparison 
with  a  chronometer  whose  correction  is  known.  Let  cx  and 
c2  be  the  two  corrections,  t  the  interval  expressed  in  days ; 
then  we  have  for  the  daily  change, 

L  v-^f*'  <119> 

that  is,  the  daily  change  is  equal  to  the  difference  of  the  two 
chronometer  corrections  divided  by  the  number  of  days  and 
parts  in  the  interval.  If  attention  is  paid  to  the  signs,  + 
will  indicate  that  the  chronometer  is  losing,  —  that  it  is  gain- 
ing. 

Examples. 


Chbo.  1615. 

Chbo.  4872. 

Chbo.  796. 

A                         h      m      s 

h      m      s 

h    m      s 

Chro.  cor.  April  15  0            +  0  18  16.2 

—  1  15  27.5 

+  00  16.6 

"       "         "      27  8            +0  18  29.6 

-  1  14  58.6 

-0  0    5.3 

Change  in  12.3  days,         +  13.4 

+  28.9 

—  21.9 

Daily  change  of  cor.             +1.09 

+  2.35 

-2.71 

At  fixed  observatories  an  interval  of  one  day  may  suffice. 
For  rating  sea-chronometers  by  observations  made  with  a  sex- 
tant and  artificial  horizon,  an  interval  of  from  5  to  15  days  is 
desirable. 

The  sea-rate  of  a  chronometer  is  sometimes  different  from 


146  NAVIGATION. 

its  rate  on  shore,  or  even  from  its  rate  while  on  board  ship  in 
port.  Some  chronometers  are  affected  by  magnetic  influences, 
so  that  their  rates  are  varied  by  changing  the  direction  of  the 
XII.  hour-mark  to  different  points  of  the  horizon.  All  are 
slightly  affected  by  changes  of  temperature,  as  perfect  com- 
pensation is  rarely  attainable.  The  excellence  of  a  chronom- 
eter depends  upon  the  permanence  of  its  rate.  The  rate  may 
be  large,  but  if  its  variations  are  small  the  chronometer  is 
good. 

170.  A  watch  is  often  used  for  noting  the  time  of  an  ob- 
servation. It  is  compared  with  the  chronometer  by  noting 
the  time  of  each  at  the  same  instant.  The  most  favorable 
instant  is  when  the  watch  shows  0*. 

Let  O and  W  be  these  noted  times  ;  then  A  W=  (C—W) 
is  the  reduction  of  the  watch  time  to  the  chronometer  time  for 
C=W+(C-W). 

Comparisons  should  be  made  before  and  after  the  observa- 
tion, and  the  results  interpolated  to  the  time  of  observation. 

A  practised  observer  may,  by  looking  at  the  watch  and 
counting  the  beats  of  the  chronometer,  make  the  comparison 
to  the  nearest  08.25.  It  is  better  to  take  the  mean  of  several 
comparisons  than  to  trust  to  a  single  one. 

A  mean  time  and  a  sidereal  chronometer  may  be  compared 
within  08.03  by  watching  for  the  coincidence  of  beats,  which 
occurs  at  intervals  of  3W,  for  chronometers,  which  beat  half- 
seconds. 


Examples. 

- 

Chro.  476. 

Chro.  4072. 

Chro.  1976. 

Chro.  1976. 

h      TO         s 

h     m       s 

h     m       s 

h      TO        s 

Chro. 

4  16  56.2 

3  15  17.5 

11  48  18.2 

1     0  28.5 

Watch 

1     5     0 

7  35  30 

3  16    0 

4  28     0 

C-W. 

+  3  11  56.2 

r-4  20  12.5 

-3  27  41.8 

-  3  27  31.6 

THE  CHRONOMETER  —  LONGITUDE.  147 

The  last  two  are  comparisons  of  the  watch  with  the  same 
chronometer.  Suppose  the  time  of  an  observation  as  noted 
by  the  watch  to  be  3*  37w  17s ;  for  finding  the  corresponding 
time  by  the  chronometer  we  have, 

The  change  of  O  -  W  in  1\2,         +  108.3 ; 
whence  the  change  in  lh        is  +    8 .6, 

and  the  change  in  21m.3  =  0A.35,  the  interval  between  the 
1st  comparison  and  the  observation,  -\-  3s.O ; 

or,  by  proportion,  we  have 

72m  :  21"\3  =  +  105.3  :  +  35.0 
Then,  Time  by  watch  =         3ft37m  17s 

C-  W=  -    3  27    38.8 
Time  by  chro.  =         0     9    38.2 

171.  Problem  44.  To  find  the  correction  of  a  chronom- 
eter at  a  place  whose  latitude  and  longitude  are  given. 

1st  Method.     (By  single  altitudes.) 

Observe  an  altitude,  or  set  of  altitudes,  of  the  sun  or  a 
star,  noting  the  time  by  the  chronometer,  or  a  watch  compared 
with  it. 

Find  from  the  altitude  (Prob.  37)  the  local  mean,  or  si- 
dereal, time,  as  may  be  required. 

The  "  local  time  "  —  the  "  chronometer  time,"  or 
c=  T-  C 

(Art.  135),  is  the  correction  of  the  chronometer  on  local  time. 
Applying  to  this  the  known  longitude  of  the  place  of  observa- 
tion, gives  the  correction  on  Greenwich  time. 

172.  If  an  artificial  horizon  is  used,  as  it  should  be  when 
practicable,  it  is  best  to  make  two  sets  of  observations  with 
the  roof  in  reversed  positions.     In  a.m.  observations  of  the 


148  NAVIGATION. 

sun  with  a  sextant  and  artificial  horizon,  the  lower  limb  of 
the  sun  and  the  upper  limb  of  its  image  in  the  horizon  are 
made  to  lap,  and  the  instant  of  separation  is  watched  for ; 
while  in  p.m.  observations  the  limbs  are  separated  and  ap- 
proaching, and  the  instant  of  contact  is  noted.  In  observa- 
tions of  the  upper  limb  this  is  reversed.  Even  a  good  observer 
may  estimate  the  contact  of  two  disks  differently  when  they 
are  separating  and  when  they  are  approaching.  Both  limbs, 
then,  should  be  observed. 

In  observing  altitudes  which  change  rapidly  it  is  better, 
when  circumstances  permit,  to  set  the  instrument  so  as  to 
read  exact  divisions  at  regular  intervals,  and  watch  the  in- 
stant of  contact.  A  good  observer,  with  a  sextant  and  arti- 
ficial horizon,  can  observe  the  double  altitudes  at  regular 
intervals  of  10'. 

173.  On  a  subsequent  day  repeat  this  observation,  and 
find  again  the  correction  of  the  chronometer.  The  difference 
between  these  two  corrections  divided  by  the  number  of  days 
and  parts  in  the  interval  is  the  daily  change,  as  in  Art.  169. 

It  is  important  that  both  the  observations  thus  compared 
should  be  at  nearly  the  same  altitude  and  on  the  same  side  of 
the  meridian  (when  the  sun  is  observed,  both  in  the  forenoon, 
or  both  in  the  afternoon),  and  in  general,  that  they  should  be 
made  with  the  same  instruments,  and  as  nearly  as  practicable 
under  the  same  circumstances.  Thus,  an  error  in  the  assumed 
latitude  and  constant  errors  of  the  instruments  or  the  observer 
will  affect  the  two  chronometer  corrections  nearly  alike,  but 
will  very  slightly  affect  their  difference,  and,  consequently,  the 
rate  determined  from  it  will  be  nearly  exact.  The  chronome- 
ter correction,  derived  from  single  altitudes,  may  be  erroneous 
a  few  seconds.     But  for  sea  chronometers  this  is  of  less  im- 


THE  CHRONOMETER—  LONGITUDE.  149 

portance  than  an  erroneous  determination  of  the  rate.  For 
instance,  suppose  the  determined  chronometer  correction  in 
error  4s,  and  the  daily  change  in  error  Is;  in  20  days  (Art. 
168)  the  computed  change  of  the  correction  will  be  in  error 
20s,  and  in  30  days  will  be  in  error  30s. 

174.     2d  Method.     (By  double  altitudes.) 

It  is  better  to  observe  altitudes  of  the  body  on  both  sides 
of  the  meridian,  and  as  nearly  at  the  same  altitude  as  practi- 
cable, either  on  the  same  day  or  on  two  consecutive  days. 

Altitudes  of  two  stars  also  may  be  used,  one  east,  the  other 
west  of  the  meridian. 

The  mean  of  the  two  results  is  better  than  a  determination 
from  either  alone ;  for  constant  errors  of  the  latitude,  the  in- 
strument, or  the  observer,  affect  the  two  results  in  opposite 
directions ;  that  is,  if  one  result  is  too  large,  the  other  is  too 
small,  and  by  nearly  the  same  amount. 

Examples  (Prob.  44.) 
1.    Chronometer  Correction. 


Pensa 

cola 

Navy- Yard, 

30° 

20'  30"  K,  87°  15'  21"  W. 

1898,  May  30, 

21* ;  Chro.  ] 

L876. 

T.  BY  Chro. 

Sextant  No.  2 

Art.  Hor.  No.  1. 

m      s 

3141 

22.7 

2  0    99  50  A 

.  end. 

m     s 

Chro.  cor.  (G.  m.  t. )  -  42  26     ) 
Daily  change              —         3.8  ) 

32    3.7 

23.3 

100    0 

32  27 

24 

100  10 

32  51 

23 

100  20 

O's  diam.  off  arc  +  32    8.3  \ 
on  arc  —  30  59.2  J 

33  14 

23.7 

100  30 

33  37.7 

100  40 

34    7.5 

23 

2  0    99  50B 

.  end. 

In  cor.                   -f-       34.5 

34  30.5 

23 

100    0 

34  53.5 

23.3 

100  10 

35  16.8 

23 

100  20 

Bar.  30.14 

35  39.8 

23.2 

100  30 

36    3 

100  40 

Ther.  76° 

3  32  39.07 

2  0  100  15 

3  35    5.18 

2  0  100  15 

150  NAVIGATION. 

Computation. 

h     TO        » 

T.  by  Chro.        3  32  39.07         O's  dec.  Eq.  of  t. 

Chro.  cor.           -  42  26          +  21  57  47.1     +  21.02  +  2  33.34     -  0.360 

'42.04  r  .720 

G.  m.  t.  May  31, 2  50 11                    f  59.6          16.82  — 1.02      J    .288 

.63  1    .011 

.15  +2  32.32        [  .002 


2.837  +  21  58  46.7 


Q  50°  07'  30"  {  I.  c.  +        ir.3        Ref.  -46 
-  16  11    j  S.  D.  -  W^'.^        Par.  4-06 


h      m      s 

O          /          // 

L.  ap.  t.,  May  30, 

21  03  51.75 

h  =    49  51  19 

—  Eq.  t. 

—  2  32.32 

L  =    30  20  30 

1.  sec      0.06398 

L.  in.  t.,  May  30, 

21  01  19.43 

p  =    68  01  13 

1.  cosec  0.03277 

T.  by  Chro. 

3  32  39.07 

2  s  =  148  13  02 

0Ch.  cor.  (L.m.t 

)  -  6  31  19.64 

s  =    74  06  31 

1.  cos     9.43745 

s 

-  h  =    24  15  12 

1.  sin      9.61360 

h      m       8 

9.14780 

t  =9  03  51.75 

1.  sin     9.57390 

h 

to      a 

©\s  dec. 

Eq.  of  t. 

T.  by  Chro.        3  35  05.18 

O        /          //                // 

m   $             s 

Chro.  cor.          — 

42  26 

+  21  58  46.7  +  21.02 

+2  32.32-0.360 

G.m.t.,  May  31,  2  52  41             in  0&.041       f  0.9 

-.01 

2.878 

+  21  58  47.6 

+  2  32.31 

0    50°  07' 

30"        I.  c.       +  17".3        Ref.  -  46 

+  15 

26          S.D.  +  15  48 

.5        Par.  +  06 

L.  ap.  t.,  May  30, 

h      m      s 

21  06  18 

O        /       // 

h  =  50  22  56 

—  Eq.  t. 

-2  32.31 

L  =   30  20  30 

1.  sec     0.06398 

L.  m.  t.,  May  30, 

21  03  45.69 

p  =  68  01  12 

1.  cosec  0.03277 

T.  by  Chro. 

3  35  05.18 

2  s  =148  44  38 

0  Chro.  cor.  (L.m.t.)  -6  31  19.49 

s  =  74  22  19 

1.  cos     9.43039 

s-h  s=  23  59  23 

1.  sin     9.60914 

Mean 

—6  31  19.57 

9.13628 

Red.  for  3A 

-.48 

t  =  9h  0Qm  18s 

1.  sin     9.56814 

Chro.  cor.  (L.  m.  t.)    —6  31  20.05  May  31,  0*. 


THE  CHRONOMETER  —  LONGITUDE.  151 

2.  Chronometer  Correction. 

Pensacola   Navy- Yard,  30°   20'   30"   N.,  87°  15'  21"  W. 
1898,  May  31,  Sh. 


T.  BY  Chro. 

Sextant  No  2. 

Art.  Hor.  ! 

No.  1. 

la   j 
9  24    2.7 

s 
22  8 

20 

100  40 

A  end. 

Chro.  cor.  (G.  in. 

m    s 

t.)-42  27     1 

24  25.5 

23.0 

100  30 

Daily  change 

-        3.8  i 

24  48.5 

25  12.5 
25  34.8 
25  58.2 

24.0 
22.3 
23.4 

100  20 

100  10 

100  00 

99  50 

0's  diam.  off  arc 
on  arc 

In  cor. 

+  32  12.5  I 
-  30  59.2  f 

+       36.6 

28  33.5 

23.5 

20 

97  40 

B  end. 

Bar.  30.14 

28  57 

23.5 

97  30 

Ther.  76° 

29  20.5 

29  43 

30  6 
30  29.5 

22.5 
23.0 
23.5 

97  20 
97  10 

97  00 
96  50 

9  25    0.37  2  0  100  15 

9  29  31.58  2  0    97  15 


Computation. 


h  m    s                      O's  dec.  Eq.  oft. 

T.  hy  Chro.            9  25  00.37         0     ,    „             „  m  $             s 

Chro.  cor.             -  42  27         +  21  57  47.1     +20.91  +2  33.34  -0.362 

(■  167.28  r  2.896 

G.m.t.,May31,   8  42  33                +3  02.1    \    14.64  -3.15   \    .253 

8.709            +  22  00  49.2    I        .19  +2  30.19    I    .003 


0         50  07  30     I.e.  +       18.3    Ref.-46 
—  16  10     S.D.-1548.5    Par.  +  C6 

h  =    49  51  20 

L  =    30  20  30  1.  sec.     0.06398 

h  m    s  p  =    67  59  11  1.  cosec  0.03288 

L.  ap.  t.,  May  31,  2  56  11.75       2s  =  US  11  01 

-  Eq.  t.  -  2  30.19  3  =     74  05  30  1.  cos      9.43790 

L.m.  t.,  May  31,    2  53  41.56    s-h  =    24  14  10  1.  sin      9.61331 

T.  by  Chro.  9  25  00.37  9.14807 

0  Ch.  c,  L.m.t.-6  31  18.81  t  =  2»  56™  11*. 75  9.57403 


152 


NAVIGATION. 


h  m    s  0\s-  dec. 

T.  by  Chro.           9  29  31.58  0     ,    „ 

Chro.  cor.            —  42.27  +  22  00  49.2  +20.91         + 

G.  m.  t.,  May  31,  8  47  05  in  0^.076  +  1.0 

8.785  4-  22  00  50.8                      4- 

18".3 


CD  48°  37'  30' 
4-  15'  25' 


I.e.  4- 

S.d. +15'48".5 


Eq.  oft. 

m    8  s 

2  30.19  -0.302 
-  .03 
2  30  16 

Ref .  -  48" 
Par.  4-  06". 


L.  ap.  t.,  May  31, 

—  Eq.  t. 

L.  m.  t. 

T.  by  Chro. 

Q  Ch.  cor.  L.  m.  t. 


3  00  42.67 
-  2  30.16 
2  58  12.51 
9  29  31.58 
6  31  19.07 


Mean  -  6  31  18.94 

Red.  for  —  3A  +  .48 

Chro.  cor.  L.  m.  t.  —  6  31  18.46 
May  31, 0*,  Chro.  cor.  (L.m  t.) 

Long. 
May  31,  6A,  Chro.  cor.  (G.  m.  t.) 


h  =    48  52  55 

L  =    30  20  30  1. 

p  =    67  59  11  1. 
2  s  =  147  12  36 

8  =    73  36  18  1. 

s-k  =    24  43  23  1. 

t  =  3A  00™  42s.  67        1. 

May  31,  0h. 

—  6h  31™  19*.25,  by  A.  m. 
4-  5    49    01  .4 

-  42    17.85 


sec       0.06398 
cosec.  0.03288 


cos 
sin 


9.45064 
9.62142 
9.16892 
sin       9.58446 


and  p.  m.  obs. 


3.    Table  of  Chronometer  Corrections. 
Chro.  of  1876 ;  fast  of  Greenwich  mean  time  and  gaining. 


G 

M.T. 

Chro.  Cor. 

Daily 

Ch. 

Remarks. 

h 

h  m    s 

1898, 

May  13 

-0  40  20.5 

-4.14 

0, 

a.  if.,  Key  West  Light- 
House. 

17  3 

41  26.8 

3.88 

o, 

a.  m.,  Key  West  Light- 
House. 

25  6 

41  58.3 

3.75 

O, 

A.  m.  &  p.  m.,  Pensacola 
Navy- Yard. 

316 

42  17.8 

o, 

a.  m.  &  p.  m.,  Pensacola 
Navy- Yard. 

Long.*  of  Key  West  Light-House,  81°  48'  40"  W. 
Long,  of  Pensacola  Navy- Yard,        87°  15'  21"  W. 

*  The  assumed  longitude  of  places  where  the  chronometer  is  rated 
should  be  stated. 


THE  CHRONOMETER  —  LONGITUDE.  153 

4.  Comparisons  and  Corrections  of  Chronometers. 
1898,  May  31,  6A,  G.  mean  time. 


Chro.  4375 

Chro.  9163. 

Chro.  789. 

Chro.  5165. 

h      m        s 

h     m      s 

h     m      8 

A        TO          s 

Chro. 

6  50  16.3 

5    3  29.7 

2  15  27.5 

11  59  16.8 

(1876) 

6  30    0 

6  31     0 

6  32  10 

6  33  30 

(1876)— Chro. 

— 0  20  16.3 

+  1  27  30.3 

+  4  16  42.5 

-5  25  46.8 

Cor.  of  (1876) 

-42  17.8 

—42  17.8 

-42  17.8 

—42  17.8 

Chro.  cor. 

-1     2  34.1 

-0  45  12.5 

+3  34  24.7 

-6  18    4.6 

or  +5  41  55.4 
175.    3d  Method.     (By  equal  altitudes.) 

A  heavenly  body  which  does  not  change  its  declination  is 
at  the  same  altitude  east  and  west  of  the  meridian  at  the  same 
interval  of  time  from  its  meridian  passage. 

If,  then,  such  equal  altitudes  are  observed  and  the  times 
noted  by  the  chronometer,  or  by  a  watch  and  reduced  to  the 
chronometer  (Art.  170),  the  mean  of  these  times,  or  the  middle 
time,  is  the  chronometer  time  of  the  star's  meridian  transit. 

The  corresponding  sidereal  time  is  the  star's  right  ascen- 
sion, when  the  first  observation  is  east  of  the  meridian  ;  12*  -j- 
the  right  ascension  when  the  first  observation  is  west  of  the 
meridian. 

This,  for  a  mean  time  chronometer,  may  be  converted  into 
local  mean  time  (Prob.  26)  ;  and  for  a  Greenwich  chronome- 
ter into  the  corresponding  Greenwich  time. 

Subtracting  the  chronometer  time,  we  have  the  correction 
of  the  chronometer. 

Example. 

1898,  Jan.  14,  at  Washington,  in  longitude  77°  2'  48"  W., 


equal  altitudes  of  a  Canis   Minoris   were  observed,  and  the 
times  noted  by  a  chronometer  regulated  to  Greenwich  mean 


time  ;  from  which  were  obtained  : 


7*  34'" 

00s.  34 

+  5  08 

11  .2 

12  42 

11  .54 

-19  35 

53.18 

17  06 

18.36 

2 

48.14 

17  03 

30  .22 

17  07 

56.01 

-4 

25  .79 

154  NAVIGATION. 

Mean  of  Chro.  times  (*  east)  2h  16m  35s. 65 
Mean  of  Chro.  times  ^*  west)  7  59  16  .38 
Chro.  time  of  *'s  transit  5   07    56  .01 

L.  sid.  t.  =  *'s  R.  A. 
Long. 
G.  sid.  t. 
-  S0  (Jan.  14) 

Sid.  int.  from  Jan.  14  0* 
Red.  to  m.  t.  int. 
G.  mean  time         (Jan.  14) 
Chro.  time 
Chro.  cor. 


176.  If  equal  altitudes  of  the  sun  are  observed  in  the  fore- 
noon and  afternoon  of  the  same  day,  the  mean  of  the  noted 
times  would  be  the  chronometer  time  of  apparent  noon,  were 
it  not  for  the  change  of  the  sun's  declination  between  the 
observations. 

Problem  45.  In  equal  altitudes  of  the  sun,  to  find 
the  correction  of  the  middle  time  for  the  change  of  the 
sun's  declination  in  the  interval  between  the  observa- 
tions. 

Solution.     Let 

h  =  the  sun's  true  altitude  at  each  observation, 
t  =  half  the  elapsed   apparent  time  between  the  observa- 
tions, 
T0  =  the  mean  of  the  chronometer  times  of  the  two  obser- 
vations, or  the  middle  chronometer  time, 
A  T0  =  the  correction  of  this  mean  to  reduce  to  the  chronom- 
eter time  of  apparent  noon  ; 
L  —  the  latitude  of  the  place, 
d  =  the  sun's  declination  at  local  apparent  noon, 
A(?=  the  change  of  this  declination  in  the  time  t\ 


THE  CHRONOMETER  —  LONGITUDE.  155 

then,  when  both  observations  are  on  the  same  day, 

t  -f-  A  T0  will  be  numerically  the  hour-angle  at  the  a.m.  ob- 
servation, 
t  —  AT0,  the  hour-angle  at  the  p.m.  observation, 
d  —  A  d,  the  declination  *  at  the  a.m.  observation, 
d-\-  Ad,  the  declination*  at  the  p.m.  observation. 

By  (76),  we  have  for  the  two  observations, 

sin  A=sinX  sin(d—Ad)-\-GOsLcos  (d—Ad)  cos  (<+AT0)  ) 
sinA=sinXsin((i+A^)  +  cosXcos  (<#-f  AJ)cos  (t—AT0)  ) 

But 

sin  (d ^  Ad)  =  sin  d cos  A^^  cos  d  sin  A d, 

cos  (<#  i  A  <#)  =  cos  d  cos  A  c?  =p  sin  d  sin  A  df, 
cos  (d  i  A  7JJ  =  cos  £  cos  A  2J  =f  sin  t  sin  A  T0 . 

Since  Ac?,  and  therefore  AT0,  are  very  small,  we  may  put 

cos  Ac?  =1,         sin  Ad  =  A  d  sin  1", 
cos  A  TJ  =  1,         sin  A  T0  =  15  A  T0  sin  1" ; 

A  d  being  expressed  in  seconds  of  arc,  and  A  T0  in  seconds  of 
time  ;  we  shall  then  have 

sin  (d  ^  A  d)  =  sin  d  -\-  A  d  sin  1"  cos  J, 
cos  (d ^  Ad)  =  cos  c?  =p  A d  sin  1"  sin  d, 
cos  (<J-A  7i)  =  cos   t  =p  15  A  r0  sin  1"  sin  £. 

Substituting  these  in  the  two  equations  (120),  subtracting  the 
first  from  the  second,  and  dividing  by  2  sin  1",  we  shall  have 

0  =  A  d  sin  L  cos  d  —  A  d  cos  L  sin  d  cos  £ 
+  15  A  TQ  cos  X  cos  d  sin  £. 

*  Strictly,  in  the  one  case,  A  d  should  be  the  change  of  declination 
in  the  time  t  +  AT0;  in  the  other,  the  change  in  the  time  t—AT0. 


156  NAVIGATION. 

Transposing  and  dividing  by  the  coefficient  of  A  T0,  we  find 
the  formula 

K  ni  Ac?  tan  L       Ac?  tan  d 

which  is  called  the  equation  of  equal  altitudes. 

Let 
Ahc?  =  the  hourly  change  of  declination  at  the  instant  of  ap- 
parent noon,  and  express 

t,  which  is  half  the  elapsed  apparent  time,  in  hours, 

then  A  d  =  AA  d  t, 

and  (121)  becomes 

Aft  d  Uan  L      A^jjtanj 
A Io  =  "      15  sin  *      +     15  tan*     '  (122) 

If  we  put  t  t 

^=-15iin7      *  =  15la^  <*» 

and 

CQ  =  the  chronometer  time  of  apparent  noon,  we  have 
AT0=AAhdt2inL+J3<\hdta,iid\ 

Cl-r.+Ar,  f         (124) 

In  these  formulas,  X  and  d  are  +  when  north,  A  c?  and  AA  c? 
are  -f-  when  the  sun  is  moving  toward  the  north. 

The  coefficients  is  — ,  since  t  <  12*, 
"  "        B  is  +  when  *  <  6h,  —  when  *  >  6*. 

The  computation  of  the  two  parts  of  A  T0  is  facilitated  by 
tables  of  log  A  and  log  B.  Such  tables  are  given  in  Chau- 
venet's  "  Method  of  Finding  the  Error  and  Rate  of  a  Chro- 
nometer," and  in  Bowditch,  Table  37. 

The  argument  of  these  tables  is  2  t,  or  the  elapsed  time. 
The  signs  of  A  and  B  are  given. 


THE  CHRONOMETER  —  LONGITUDE.  157 

Apply  the  two  parts  of  A  T0,  according  to  their  signs,  to 
the  Middle  Chronometer  Time;  the  result  is  the  Chronometer 
Time  of  Apparent  Noon. 

Apply  to  this  the  equation  of  time  {adding,  when  the  equa- 
tion of  time  is  additive,  to  mean  time ;  otherwise  subtracting)  ; 
the  result  is  the  Chronometer  Time  of  Mean  Noon  at  the 
place. 

Applying  to  this  the  longitude  (in  time),  subtracting  if  west, 
adding  if  east,  gives  the  Chronometer  Time  of  Mean  Noon  at 
Greenwich. 

*  12*  —  Chro.  T  at  local  Mean  Noon  will  be  the  Chro.  correc- 
tion, if  the  chronometer  is  regulated  to  local  time. 
12*  —  Chro.  T  at  Greenwich  Mean  Noon  will  be  the  Chro. 
correction,  if  the  chronometer  is  regulated  to  Greenwich 
time. 

177.  If  a  set  of  altitudes  is  observed  in  the  afternoon  of 
one  day,  and  a  set  of  equal  altitudes  in  the  forenoon  of  the 
next  day,  the  middle  time  would  correspond  nearly  to  the  in- 
stant of  apparent  midnight ;  and  half  the  elapsed  time  t  would 
be  nearly  the  hour-angle  from  the  lower  branch  of  the  merid- 
ian, or  the  supplement  of  the  proper  hour-angle. 

In  this  case 

180°  —  (t  -f  A  T0)  will  be  the  hour-angle  at  the  p.m.  observation. 
180°  —  (t  —  A  T0)    "     "    «  «  "  "    a.m.         " 

d  —  Ad,  the  declination  at  the  p.m. 

d  +  A  d,  "  "  "     "  a.m. 

and  we  have  for  the  two  observations,  as  in  (120) 
*  This  is  better  noted  as  0h. 


ti 


sin  A=sinZ,  s'm(d—&d)  —cosLcos(d—Ad)  cos(£-f  A  T0) 


158  NAVIGATION. 

smh=smLsm(d-\-Ad)—cosLcos(d-\-Ad)cos(t—AT0))    v      ' 
Treating  these  in  the  same  way  as  (120)  we  shall  have 

0  =  A  d sin  L  cos  d  -f-  A  <#cos  L  sin  d cos  t 

—  15  A  T0  cos  L  cos  c?  sin  £ ; 
whence 

.    yr A(?  tan  L       £±d  tan  ^ 

0  "      15  sin  t  15  tan  t 

or,  putting  as  before  A  d  =  Ah  d  t 

A  = - ,       B 


15  sin  t'  15  tan  t9 

&  TQ=  —  A  &hd  ten  L  +  JB  &hd  tan  d,  (126) 

which  differs  from  (124)  only  in  the  sign  of  A.  This  is  the 
reduction  of  the  middle  time  to  the  Chro.  Time  of  apparent 
midnight:  applying  the  equation  of  time  reduces  it  to  the 
Chro.  Time  of  mean  midnight. 

178.  d,  A  dy  and  the  equation  of  time,  are  to  be  taken 
from  page  I  of  the  Almanac,  and  interpolated  as  in  Art.  90 
for  the  instant  of  apparent  noon,  or  of  apparent  midnight, 
according  as  the  observations  are  made  on  the  same  day, 
or  on  consecutive  days. 

2  t  is  properly  the  elapsed  apparent  time.      The  elapsed 

*  These  may  be  written 

—  sin  h  =  —  sin  L  sin  (d  —  A  d)  +  cos  L  cos  (d  —  A  d)  cos  (t  +  A  ^o) 

—  sin/i  =  —  sin  L  sin  (d  +  \d)  +  cos  Lcos  (d  +  A^)cos(£  —  A  T0). 

They  differ  from  (120)  in  the  signs  of  h  and  X,  and  in  reckoning  the 
hour-angles  from  the  lower,  instead  of  the  upper,  branch  of  the  meridian. 
This  would  be  the  case,  if  we  suppose  the  observations  to  be  referred  to 
the  latitude  and  meridian  of  the  antipode.  The  only  effect  in  (121)  is 
to  change  the  sign  of  tan  L,  or  of  the  first  term  in  the  equation  of  equal 
altitudes. 


THE  CHRONOMETER  —  LONGITUDE.  159 

time  by  chronometer  requires,  then,  not  only  a  correction 
for  the  rate,  which  is 

2  t 

A  c  (+  when  the  chronometer  loses)  ;  (41) 

but  also  a  reduction  to  an  apparent  time  interval,  which, 
for  a  mean  time  chronometer,  is  the  change  *  of  the  equation 
of  time  in  the  time,  2  t,  additive  when  the  equation  of  time 
is  additive  to  mean  time  and  increasing,  or  subtractive  from 
mean  time  and  decreasing.  For  a  sidereal  chronometer,  it 
is  the  change  in  the  sun's  right  ascension  in  the  time  2 1, 
and  subtractive. 

179.  Equal  altitudes  of  the  moon  or  a  planet  may  be  ob- 
served ;  but  in  the  case  of  the  moon  admit  of  less  precision 
than  of  the  sun,  and  moreover  require  correction  for  the 
inequality  produced  by  change  of  parallax. 

If  2  A  a  is  the  increase  of  right  ascension  in  the  interval, 
the  body  will  arrive  at  its  second  position  later  than  would 
a  fixed  star,  supposed  coincident  with  it  at  the  first  posi- 
tion ;  and  the  elapsed  sidereal  time  will  be  greater  than  the 
double  hour-angle  of  the  body  by  the  quantity  2  A  a.  If 
2  s  =  the  elapsed  sidereal  time,  then  in  (122)  we  must  take 

2t  =  2s  —  2Aa,OYt  =  s  —  Aa.  (127) 

If  t  m  =  half  the  elapsed  mean  time  (expressed  in  hours  when 

used  as  a  coefficient),  and 
AA  a  =  the   increase  of   right  ascension  in  lh  of  mean  time, 

by  (64)  s  =  tm  +  9S.8565  tm 

and  t  =  tm  +  tm  (9S.8565  —  A*  a),  (128) 

*  The  maximum  daily  change  is  30s.  The  elapsed  time  by  Chro- 
nometer is  usually  regarded  as  sufficiently  accurate. 


160  NAVIGATION. 

by  which  t  and  2  t  may  be  found  from  2  tm  the  elapsed  mean 
time. 

In  this    expression    the   last   two   terms    are    in   seconds. 
Reducing  to  hours  we  have 

t  =  tm  (l  +  9*-8565  -**«)  =  tj  1.002738  -  ^)   (129) 
WV  3600  J        m\  3600  J  V      ; 

If  AA  d  =  the  change  of  declination  in  lh  of  mean  time,  then 

in  (121) 

Adf=  tmAhd 

or,  substituting  for  tm  its  value  from  (129), 


A  d  =  t  Ahd 


■s-  f  1.002738  -  -^Y 
V  3600/ 


Equations  (123)  and  (124)  may  then  be  used  for  other 
bodies  than  the  sun,  provided  we  give  t  its  proper  value  from 
(127)  or  (128),  and  for  Ahd  substitute 

A',  d=Ahd+(  1.002738  -  **1*\ 


3600  y' 

or,  which  will  be  sufficiently  exact, 

w    7       k     j  .   Aha  —  98.856  A     7  /--u_v 

A'*  df  =  A*  <*  +     h    36Q() AA  £  (130) 

180.  Observing  the  double  altitudes  at  regular  intervals 
of  10',  or  20',  especially  facilitates  the  method  of  equal  alti- 
tudes; for,  if  the  first  set  is  observed  at  equal  intervals,  in 
the  second  the  observer,  having  set  the  instrument  for  the 
last  reading  of  the  first  and  observed  the  contact,  for  the 
subsequent  observations  has  only  to  move  back  successively 
the  same  intervals. 

181.  It  is  not  requisite  that  the  instrument  should  give 
the  true  altitude ;  it  is  sufficient  if  the  altitude  is  the  same 


THE  CHRONOMETER — LONGITUDE.  161 

at  the  two  corresponding  observations.  Hence  the  two  obser- 
vations should  be  made  with  the  same  instruments,  without 
change  of  adjustment,  and  in  general  as  nearly  as  practicable 
under  the  same  circumstances. 

This  purpose  is  promoted  by  making  the  final  movement 
of  the  tangent  screw  in  both  sets  always  in  the  same  direc- 
tion. Thus,  in  reversing  the  movement,  the  screw  may  be 
turned  a  little  too  far,  and  then  the  final  contact  made  by 
a  motion  in  the  same  direction  as  before. 

If  the  sun  is  used,  both  limbs  should  be  observed. 

The  error  arising  from  want  of  parallelism  of  the  surfaces 
of  the  roof-glasses  of  the  horizon  is  eliminated  by  having  the 
same  end  of  the  roof  toward  the  observer.  The  roof  may 
be  tested  by  observing  sets  of  altitudes  with  it  in  reversed 
positions. 

182.  Although  the  readings  of  the  instrument  may  be 
the  same  in  the  two  sets  of  observations,  the  altitudes  may 
be  slightly  different,  1st,  from  changes  in  the  instrument  in 
the  interval ;  2d,  from  difference  of  refraction  at  the  two 
times. 

A  change  in  the  index  correction  may  be  detected  by 
observation ;  but  there  may  be  expansion  or  contraction  of 
various  parts  of  the  instrument  which  may  affect  the 
readings  of  the  altitudes  without  altering  the  index  correc- 
tion. 

The  change  of  refraction  may  be  found  by  noting  the 
barometer  and  thermometer  at  each  set,  and  finding  the  re- 
fraction for  both  sets  of  altitudes. 

183.  To  correct  the  middle  time  for  any  small  difference 
of  the  altitudes,  whether  from  refraction  or  actual  change  of 


162 


NAVIGATION. 


readings,  we  may  find,  from  the  difference  between  two  read- 
ings, and  the  difference  of  the  corresponding  times,  the  change 
of  time  for  a  change  of  1',  or  1",  of  altitude.  This  multiplied 
by  half  the  inequality  of  altitudes,  expressed  in  minutes,  or 
seconds,  will  give  the  correction  of  the  middle  time,  to  be 
added  when  the  p.m.  altitude  is  the  greater ;  to  be  subtracted 
when  the  p.m.  altitude  is  the  less. 

If  twice  the  altitude  is  observed  with  an  artificial  horizon, 
we  may  find  the  change  of  time  for  a  change  of  1',  or  1",  of 
the  double  altitude,  and  multiply  it  by  the  whole  inequality 
of  the  altitudes. 


Example.     (Prob.  45.) 

1.    1898,  Jan.  10,  9fc*  a.m.  and  2j*  p.m.     Equal  altitudes  of 
O  at  the  Custom  House,  Key  West,  Florida ; 
24°  33'  20"  K,  81°  48'  37"  W.     Chro.  1085 ; 
Chro.  cor.  (G.  m.  t.) 


42m  18s.O  ;  daily  change  +  8  3. 


Sex.  No.  1. 
Art.  Hor.  No.  2. 

o  / 

2  0  GO   0  A.  end. 

10 

20 

30 

40 

50 
2060   0 

10 

20 

30 

40 

50 


60  25 


T.  by  Chro. 


a.m. 

h  m   s 
3  39  26.7 

39  58.8 

40  31.5 

41  4.0 

41  35.7 

42  9.0 

42  58.5 

43  31.3 

44  3.8 

44  37.5 

45  10.0 
45  43.7 

3  42  34.21     8  52  51.46 


P.M. 

h  m    8 

8  55  59.7 
55  27.0 
54  55.3 
54  22.0 
53  50.3 
53  16.5 
52  26.7 
51  53.5 
51  21.0 
50  48.0 
50  15.5 
49  42.0 


Mid.  Time. 
6A17OT 

8 

43.2 
42.9 
43.4 
43.0 
43.0 
42.8 
42.6 
42.4 
42.4 
42.7 
42.8 
42.8 


Elapsed  Chro.  t. 
Mid.  Chro.  t. 
1st  part  of  Eq. 
2d  part  of  Eq. 
Chro.  t.  of  ap.  noon 
Eq.  t. 


5  10  17.25 

6  17  42.83 

—  2.89 

—  1.99 
6  17  37.95 
—  7  57.18 


Long. 


O's  diam.  H 

In  cor. 

Bar. 
Ther. 

Ref.     - 


32'  25".0 

32  41   .7 

-8.3 

30.22 

77° 

1'  36" 


p.m. 

+  32'26".7 
-32  43  .3 

-8.3 

30.18 


-1' 


h    TO      8  m    8  8 

+  5  27  14.5        Eq.  t.    —7  51.74—0.997 

5.456  z-4.985 

0d.  227  —5.44  J    .399 

J    .050 

—  7  57.18    I  .006 


THE  CHRONOMETER — LONGITUDE.  163 


Chro.  t.  of  mean  noon    6  09  40.77         0's  dec.  —21  54  46.4  Ah  d  +22.78    Ch.  inld+1.06 

Long.  (W.)                     —5  27  14.5  +.24                    ,.212 

Chro.  t.  of  G.  m.  noon      0  42  26.27  +23.02                    J  .021 

Chro.  cor.  (G.  m.  t.)     —    42  26.27  22.90                    '  .007 

Jan.  10,  5^.5  fl  14.50 

+  2  05.0  J      9.26 

1      1.15 

—  2152  41.4  I       .14 

L  -  +  24°  33'  20"  1.  tan  9.6598  d  =  -  21°  54'  46"  1.  tan  9.6045  n 

Ahd=  +23"  .02  log       1.3621  log       1.3621 

A  log      9.4396  m  B  log      9.3315 

—  2s.89  log      0.4615  n  —  ls.99  log      0.2981  n 

(The  elapsed  apparent  time  is  5h  10™  12s.) 

2.  1898,  June  19,  4J*  p.m.,  and  20,  1\  a.m.  ;  nearly  equal 
altitude  of  Q;  at  Belize,  S.  E.  pass  of  Mississippi  River, 
29°  V  8"  K,  89°  5'  18"  W.,  chro.  1085;  chro.  cor.  (G.  m.  t.) 
—  41m  28s ;  daily  change  -f  15.0 ;  sextant  No.  2 ;  art.  hor. 
No.  1;  (A.  end  toward  observer). 


P.  M  A.M.  P.M.  A.M. 

Sex.  Read.       T.  by  Chr.       T.  by  Chr.      Sex.  Read.     In.  cor.  +  41//.8        +  40" .4 


o         / 

h     m    s 

ft  m    8 

0        > 

2  ©65  10 

10  55  48 

2  22  44.5 

2Q65  30 

Bar.                    30  .09         30   .12 

65   0 

56  10.8 
56  34.3 

22  21 
2134 

65  20 
65   0 

Ther.                       81°                 80° 

64  50 

J  In  cor.     +    '  20".9   +  '  20". 2 

64  40 

56  57.5 

21  11.2 

64  50 

Ref.             -1   28         -1   28 

57  27.5 

20  41.5 

2  0  65  40 

-  1   07   .1    -1  07  .8 

2  065  30 

(P.M.  — A.M.) 

65  20 

57  50.5 

20  18 

65  30 

Diff.  ref.,  etc.              +    /   0".7 

65  10 

58  14.3 

19  54 

65  20 

Diff.  obs.  alts.             —5  37  .5 

65    0 

58  37.3 

19    7.5 

65    0 

A/i  =  -5  36  .8 

2  0  65    5  0        10  57  12.53         2  20  58.96        2  0  65  16  15  For  2  A  h  =  10',     A  t  =  23s.3 

/i'=32  32  30  fc'=32  38    7.5  2  A  h  =    1'      At=    2s.38 


G.  ap.  t.,  June  19,  17*  56m  21s.2  =  19,  17A.939  =  19*747 


O's  Dec. 

Ah  d 

Ch.  in  Id. 

Eq.  of  T. 

+  23°  26'  25".4 

+  1".99 

-  1".04     1.60 

—  lm  04s.39  —  0».546 

f      -73 

r  5.46 

+  28   .7 

-.78 

\      .04 
I      .01 

-9.80    - 

3.82 
.50 

-f-  23   26  54 

+  1.21 

-  1     14.19 

{     .02 

164 

naviga: 

riON. 

h     m      8 

h    m      8 

Middle  chro.  t. 

18  39  05.75 

Elap.  t.  by  chro.*         15  23  46 

Red.  for  A  h,—  5'. 61  x  2s. 

33       - 13.07 

-26 

1st  part  of  Eq. 

+      .38 

Ch.  of  Eq.  t. 

-    8 

2d  part  of  Eq. 

-      .13 

Elap.  ap.  t. 

15  23  12 

Chro.  t.  of  ap.  12* 

18  38  52.93 

O        /        // 

Eq.  of  t. 

-    1  14.19 

L=  +29  07  08 

1.  tan  9.7459 

Chro.  t.  of  mean  12* 

18  37  38.74 

&hd  =  +  l".21 

log  0.0828 

Long. 

—  5  56  21.2 

A 

log  9.7542 

Chro.  t.  of  G.  mean  12* 

12  41  17.54 

+0S.38 

log  9.5829 

Chro.  cor.  (G.  m.  t.) 

-  41  17.54  June  19,  18*. 

d 

=  -  23°  26'  54"      1. 

tan  9.6372 
log  0.0828 

B 

log  9.3862  n 

—  0s.  13 

log  9.1062  n 

184.  4th  Method  of  finding  the  correction  of  a  chronom- 
eter.    (By  transits.) 

On  shore  the  most  accurate  method  of  finding  the  correc- 
tion of  a  chronometer  is  by  noting  the  times  of  transit  of  the 
sun  or  a  star  across  the  threads  of  a  well-adjusted  transit 
instrument.  The  mean  of  these  times  is  taken  and  corrected 
for  the  errors  of  the  instrument,  or  reduced  to  the  meridian. 
In  the  case  of  the  sun,  the  transits  of  both  limbs  may  be 
observed ;  or  only  one,  and  the  "  sidereal  time  of  the  semi- 
diameter  passing  the  meridian,"  found  on  page  I  of  each 
month  in  the  almanac,  added  for  the  limb,  which  transits 
first ;  subtracted  for  the  second  limb. 

At  the  instant  of  a  star's  transit  of  the  meridian,  the  right 
ascension  of  the  star  is  the  sidereal  time.  The  instant  of 
transit  of  the  sun's  centre  is  apparent  noon. 

*  Twice  the  reduction  of  the  middle  time  for  the  diff.  of  alts,  is  to  be 
added  to  the  elapsed  time  when  the  p.m.  observation  is  last  ;  subtracted 
when  the  p.m.  observation  is  first.  This  may  be  neglected  unless  the 
diff.  of  altitudes  is  quite  large. 


THE  CHRONOMETER  —  LONGITUDE.  165 

From  either  of  these,  the  local  sidereal  or  mean  time,  as 
may  be  required,  can  be  found ;  and  thence  the  chronometer 
correction  by  subtracting  the  chronometer  time  of  transit. 

The  moon  should  not  be  used  for  finding  the  time,  when 
precision  is  required.  Stars  are  preferred  to  the  sun,  either 
when  transits  are  observed,  or  equal  altitudes  with  the  arti- 
ficial horizon  ;  chiefly  because  many  stars  may  be  observed 
during  the  same  night,  and  the  instrument  is  not  exposed  to 
the  rays  of  the  sun. 

185.  By  repeating  the  transits  on  a  subsequent  day,  the 
chronometer  correction  can  be  again  found,  and  from  the  two 
corrections,  the  rate,  as  in  Art.  169.  If  the  transit  instru- 
ment is  not  well  adjusted,  or  the  instrumental  corrections  are 
imperfectly  known,  the  rate  of  the  chronometer  can  still  be 
quite  well  determined  from  transits  of  the  same  star,  or  the 
same  set  of  stars,  on  different  days,  provided  the  position  of 
the  instrument,  or  its  adjustments,  have  not  been  disturbed  in 
the  interval. 

186.  Fifth  Method.  (By  time  signals)  These  signals  can 
be  obtained  at  the  W.  U.  Telegraph  Office  in  any  part  of  the 
United  States,  without  delay,  in  any  weather,  and  with  abso- 
lute certainty  as  to  comparisons.  Since  well-adjusted  transit 
instruments  are  not  generally  available,  the  electric  signal 
from  Washington  furnishes  the  best  means  of  rating  chro- 
nometers, with  the  utmost  simplicity  of  method,  and  a  high 
degree  of  accuracy.  But  if  time  signals,  or  a  time-ball,  be 
employed,  the  chronometer  error  should  be  found  also  by 
astronomical  observations,  as  a  check ;  for  in  so  important  a 
matter,  the  navigator  ought  not  to  accept  the  unsupported  or 
uncorroborated  work  of  another  person.  ("  Notes  on  Naviga- 
tion."    Nav.  Academy,  1872.) 


166  NAVIGATION. 

LONGITUDE. 

187.  To  find  the  longitude  of  a  place  by  astronomical  ob- 
servations, it  is  generally  necessary  to  determine  indepen- 
dently the  local  and  Greenwich  times  of  the  same  instant.  The 
difference  of  these  times  is  the  longitude,  which  is  west  when 
the  Greenwich  time  is  the  greater,  and  east  when  the  Green- 
wich time  is  the  less  (Art.  165).     This  is  expressed  by  (50) 

x  =  T„  -  T, 

in  which  T0  is  the  Greenwich  time,  and 

Ty  the  corresponding  local  time  of  the  same  kind. 

These  times  may  be  apparent,  mean,  or  sidereal. 

The  apparent  time  is  the  hour-angle  of  the  true  sun ;  the 
mean  time,  that  of  the  mean  sun ;  the  sidereal  time,  that  of 
the  vernal  equinox.  In  the  same  way  we  may  use  the  local 
and  Greenwich  hour-angles  of  any  other  body  or  point  of  the 
heavens,  regarded  as  +  toward  the  west. 

This  is  evident  from  Fig.  35 ; 
for  if 

P  M  is  the  meridian  of  Greenwich, 
P  M',  the  local  meridian, 
P  S,  the  declination  circle  of   a 

heavenly  body ; 
MPM'  will  be  the  longitude  of 
the  place, 

MPS,  the  hour-angle  of  the  body  at  Greenwich, 
M'  P  S,  the  local  hour-angle  ; 
and  we  shall  have,  as  in  Art.  74, 

MPM'  =  MPS-M'PS. 

The  several  methods  of  finding  the  longitude  differ  in  the 
modes  of  finding  and  comparing  the  two  times,  or  the  two 
hour-angles. 


LONGITUDE.  167 

188.  Problem  46.  To  find  the  longitude  of  a  place  by 
a  portable  chronometer  regulated  to  Greenwich  time. 

Solution.  The  correction  and  rate  of  the  chronometer  are 
supposed  to  have  been  found  by  suitable  observations  at  a 
place  whose  longitude  is  known.  Let  the  chronometer  be 
transported  to  the  place  whose  longitude  is  required ;  and  let 
an  observation  suitable  for  finding  the  hour-angle  of  a  heavenly 
body,  or  the  local  time,  be  made,  and  the  time  noted  by  the 
chronometer,  or  by  a  watch  compared  with  it. 

There  are  then  two  parts  of  the  process  to  be  pursued  : 
1st,  from  the  noted  time  to  find  the  Greenwich  time  (mean, 
apparent,  or  sidereal),  or  the  hour-angle  of  the  body,  as  may 
be  deemed  most  convenient.  2d,  from  the  observations,  to 
find  the  corresponding  local  time,  or  hour-angle.  Subtracting 
the  local  time,  or  hour-angle,  from  the  Greenwich  time,  or 
hour-angle,  will  give  the  longitude. 

189.  1st.  To  find  the  Greenwich  time,  or  hour-angle,  of 
the  body  observed,  apply  to  the  noted  time  the  reduction  of  the 
watch  time  to  chronometer  time,  C  —  W  (if  a  watch  has  been 
used)  and  the  chronometer  correction,  c' ,  reduced  to  the  date 
of  observation  (Art.  168). 

The  result  is,  the  Greenwich  time ;  and  will  be  mean  or 
sidereal,  according  as  the  chronometer  is  regulated  to  mean 
or  sidereal  time.*  If  it  is  sidereal  time,  it  will  be  necessary 
to  reduce  it  to  mean  time  (Prob.  26),  except  when  a  fixed 
star  has  been  observed,  so  as  to  take  from  the  Almanac  the 
quantities  which  will  be  required. 

If,  now,  the  Greenwich  hour-angle  of  the  body  observed  is 
desired : 

*  For  observations  of  stars,  a  sidereal  chronometer  is  most  conve- 
nient. 


168  NAVIGATION. 

In  the  case  of  the  sun,  reduce  the  Greenwich  mean  time  to 
apparent  time,  by  applying  the  equation  of  time. 

If  some  other  body  has  been  observed,  reduce  the  Greenwich 
mean  time  to  sidereal  time  by  adding  the  right  ascension  of 
the  mean  sun ;  and  thence  find  the  hour-angle  of  the  body  by 
subtracting  its  right  ascension.  Or,  if  a  sidereal  chronometer 
has  been  used,  from  the  Greenwich  sidereal  time  subtract  the 
right  ascension  of  the  body. 

Attention  to  the  signs  will  give  the  hour-angle  thus  ob- 
tained, -f-  if  toward  the  west,  —  if  toward  the  east. 

190.  The  Greenwich  time,  or  hour-angle,  is  affected  by  the 
error  of  the  chronometer  correction,  which  consists,  1st,  of  the 
error  in  its  original  determination,  which  includes  any  error 
of  the  assumed  longitude  of  the  place  of  rating;  2d,  of  the 
error  arising  from  an  erroneous  rate.  This  last  error  is  cu- 
mulative, increasing  with  the  number  of  days  from  the  date, 
when  the  correction  of  the  chronometer  was  found  from  obser- 
vations. 

191.  The  chronometer  correction  for  the  date  of  observation 
can  be  derived  from  subsequent  as  well  as  from  prior  deter- 
minations of  it  and  its  daily  change.  In  finding  the  longitude 
of  a  place  on  shore,  or  of  a  shoal,  both  values  should  be  ob- 
tained, when  practicable,  and  combined  by  giving  weights  to 
each  inversely  proportional  to  its  interval  of  time  from  the 
original  determination.  Thus,  if  cf  and  c"  are  two  such  chro- 
nometer corrections,  the  first  brought  forward  t'  days,  the  sec- 
ond carried  back  if'  days,  we  may  take  as  the  mean  value  * 

*  This  assumes  that  c'  and  c"  are  derived  from  two  chronometer 
corrections  of  equal  weight,  and  consequently  that  the  longitudes  used 
in  finding  them  are  equally  reliable.     This  may  not  be  the  case  if  the 


LONGITUDE.  169 

f  d  +  1  c" 

a  + 1"    ' 

or,  in  a  form  more  convenient  for  computation, 

For  example,  suppose  that  on  Jan.  17,  the  chronometer  correc- 
tion brought  forward  from  Jan.  1,  is  —  18m  56s.5,  and  reduced 
back  from  Jan.  25,  is  —  19m  35.4 ;  the  value  by  the  above 
formula  will  be 

-  18-  56s.5  +  16  X  ~  68,9  =  -  19m  1M. 

Two  longitudes  may  be  combined  in  a  similar  way. 

192.  Reports  of  longitudes  by  chronometer  are  regarded  as 
of  but  little  value,  unless  the  number  of  chronometers,  the  as- 
sumed longitude  of  the  place  where  the  chronometer  is  rated, 
and  the  age  of  the  rates,  are  stated.  Strictly,  the  chronometer 
merely  determines  the  difference  of  longitude  between  the  two 
places  where  the  observations  are  made.  This  may  be  ob- 
tained by  using  the  chronometer  correction  on  the  time  of  the 
place  of  rating,  instead  of  the  Greenwich  time.  It  is  prefera- 
ble to  report  such  differences  rather  than  absolute  longitudes. 

193.  2d.  To  find  the  hour-angle  of  the  body,  and  thence 
the  local  time. 

1st  Method.  (Prob.  37.  By  single  altitudes.)  Observe 
in  quick  succession  several  altitudes  of  the  heavenly  body, 

chronometer  corrections  were  found  from  observations  at  two  different 
places. 

The  student  is  referred  to  Chauvenet's  "  Astronomy,"  I.,  317,  etc., 
for  the  methods  of  allowing  for  changes  in  the  rates  and  combining  the 
results  of  several  chronometers. 


170  NAVIGATION. 

noting  the  time  of  each  by  the  chronometer,  or  by  a  watch 
compared  with  it. 

Take  the  mean  of  the  noted  times,  and  from  it  find  the 
Greenwich  mean  time;  for  which  take  from  the  Almanac 
the  declination  of  the  body,  its  semidiameter  and  horizontal 
parallax  when  sensible,  as  well  as  the  quantities  required  for 
finding  the  Greenwich  hour-angle.     (Art.  189.) 

Take  the  mean  of  the  readings  of  the  instrument,  with 
which  the  altitudes  were  measured,  and  from  it  find  the  true 
altitude  of  the  centre  of  the  body.  (Art.  118.)  With  this 
and  the  known,  or  assumed,  latitude  of  the  place  find  the 
local  hour-angle  of  the  body  by  Problem  37. 

This  hour  angle,  which  for  the  sun  is  the  local  apparent 
time,  subtracted  from  the  corresponding  Greenwich  hour- 
angle  already  found,  will  give  the  longitude. 

Or,  the  local  mean  time  may  be  found  from  it,  for  the  sun, 
by  applying  the  equation  of  time;  for  other  bodies,  by  add- 
ing the  right  ascension  of  the  body,  which  will  give  the 
local  sidereal  time,  and  subtracting  the  right  ascension  of  the 
mean  sun  (Prob.  31)  :  and  the  local  time  subtracted  from 
the  corresponding  Greenwich  time  will  give  the  longitude. 

194.  On  shore  it  is  best  to  use  an  artificial  horizon,  even 
when  a  sea-horizon  can  be  had,  and  for  precise  observations, 
stars  in  preference  to  the  sun. 

At  sea  the  sun  is  most  conveniently  used  ;  but  altitudes 
of  the  moon  and  bright  stars  can  be  employed  when  the 
sun  is  not  available.  The  chief  difficulty  is  the  obscurity  of 
the  sea-horizon  at  night.  During  twilight,  however,  or  in  a 
bright  moonlight,  it  is  often  distinct  and  well  defined. 

195.  The  most  favorable  position  of  the  body  for  finding 


LONGITUDE.  171 

its  hour-angle  from  its  altitude  is,  as  previously  stated,  when 
it  is  nearest  the  prime  vertical ;  provided  its  altitude  is  not 
so  small  as  to  involve  to  too  great  an  extent  the  uncertainty 
of  refraction ;  and,  observed  on  shore,  is  within  the  limits  * 
of  the  instruments  employed. 

•  On  shore  the  time  and  circumstances  most  favorable  for 
observations  can  generally  be  selected.  At  sea  long  con- 
tinuance of  bad  weather  may  render  poor  observations,  made 
under  unfavorable  circumstances,  the  only  ones  available. 

While,  then,  it  is  not  well  to  use  for  finding  the  time 
an  altitude  less  than  10°,  or  of  an  object  whose  azimuth  is 
less  than  45°  or  more  than  135°,  it  may  sometimes  be  neces- 
sary to  exceed  these  limits. 

196.  When  the  declination  and  latitude  are  nearly  the 
same,  the  body  is  nearest  the  prime  vertical  but  a  short  time 
before  and  after  its  meridian  passage,  so  that  a  very  great 
altitude  may  be  used.  Thus  in  lat.  20°  N.,  the  sun,  when  its 
declination  is  19°  55'  N.  or  20°  5'  N.,  is  nearest  the  prime 
vertical  within  22m  of  noon  at  an  altitude  of  nearly  85° ; 
and  the  local  time  can  be  as  accurately  obtained  from  an 
altitude  of  89°,  4TO  from  noon,  and  about  5°  in  azimuth  from 
the  prime  vertical,  as  from  an  altitude  of  30°,  provided  the 
assumed  latitude  can  be  depended  on  within  2'.  Nearer 
noon,  the  rapid  change  of  the  sun's  azimuth,  averaging  10° 
in  lm,  would  make  it  difficult  to  observe  the  altitude  with 
sufficient  precision. 

197.  The  local  time  or  hour-angle  is  affected  by  errors  in 
the  altitude  and  in  the  assumed  latitude.  (Arts.  136,  138.) 
When  several  observations  have  been  made  in  rapid  succession, 

*  For  a  sextant  and  artificial  horizon,  between  20°  and  60°. 


172  NAVIGATION. 

the  effect  of  a  supposed  error  of  V  in  the  altitude*  may- 
be found  by  dividing  the  difference  of  two  of  the  noted  times 
by  the  difference,  in  minutes,  of  the  corresponding  altitudes. 

In  a  similar  way  we  may  find  the  change  of  altitude  in 
l9*  of  time  by  dividing  the  difference  of  two  altitudes  by 
the  difference  in  minutes  of  the  corresponding  times.  The 
maximum  change  of  altitude  in  lm  is  15';  when  L  =  0  and 
d  =  0.  The  more  rapid  the  change  of  altitude,  the  less  will 
errors  of  altitude  affect  the  result. 

To  ascertain  the  effect  of  an  error  of  1'  in  the  assumed 
latitude,  f  the  local  times  or  hour-angles  may  be  computed 
separately  for  two  latitudes  differing  10',  or  20',  from  each 
other,  and  the  difference  of  these  times  divided  by  10',  or 
20'.  At  sea  the  latitude  by  account  is  used,  either  brought 
forward  to  the  time  of  observation  from  a  preceding,  or  car- 
ried back  from  a  subsequent,  determination.  It  may  be  very 
largely  in  error,  especially  in  uncertain  currents,  or  after  run- 
ning several  days  without  observations. 

A  small  error  may  also  result  from  the  assumption  that 

*  Differentiating  equation  (76) 

sin  h  —  sin  L  sin  d  +  cos  L  cos  d  cos  £, 
regarding  h  and  t  as  variables,  we  have 

cos  h  d  h  =  —  cos  L  cos  d  sin  t  d  t 
but  cos  d  sin  t  =  cos  h  sin  Z  Sph.  Trig.  (114) 

whence  d  t  =  —  — — ; — - » 

15  cos  L  sin  Z 

which  is  a  minimum  when  Z  =  ±  90°,  and  incalculable  when  Z  =  0° 
or  180°. 

t  From  (96)  we  find 


&t=  - 


15  cos  L  tan  Z 
which  is  0,  when  Z—  ±  90°,  and  also  incalculable  when  Z  =  0  or  180°. 


LONGITUDE.  173 

the  mean  of  the  instrumental  readings  corresponds  to  the 
mean  of  the  noted  times.  The  reduction  of  the  mean  of  the 
altitudes  to  the  mean  of  the  times  can  be  found,*  but  it  can 
be  avoided  by  limiting  the  series  of  observations,  which  are 
combined  together,  to  so  brief  a  period  that  the  error  becomes 
insensible ;  or,  when  the  body  is  near  the  meridian  in  azimuth, 
by  reducing  each  observation  by  itself.  This  last  case,  how- 
ever, should  be  avoided  in  this  problem. 

198.  At  sea  it  is  usual  to  reduce  longitudes  obtained  from 
day  observations  to  noon  by  allowing  for  the  run  of  the  ship 
in  the  interval,  and  for  currents  when  known.  Those  from 
night  observations  are  recorded  for  the  time  of  observation. 

199.  2d  Method.  Altitudes  in  the  forenoon  and  in  the  af- 
ternoon, or  on  different  sides  of  the  meridian,  are  preferable 
to  single  altitudes  for  finding  the  local  time,  for  the  reasons 
already  stated  in  Article  174.  The  longitudes  can  be  found 
from  each  set  separately,  and  then  combined. 

At  sea  the  longitudes  derived  from  each  can  be  reduced  to 
noon,  and  the  mean  of  the  two  taken  as  the  true  longitude  ; 
or,  if  the  difference  can  be  regarded  as  due  to  currents,  the 
longitude  at  noon  can  be  found  by  interpolating  for  the  elapsed 
time.  It  is  desirable  that  the  observations  should  be  made  at 
nearly  equal  intervals  from  noon. 

Longitudes  by  a.m.  and  p.m.  observations  are  enjoined  in 
the  directions  of  the  Navy  Department  whenever  practicable. 

Example.     (Prob.  46.) 

1.  At  sea,  May  17, 1898,  9*  45™  a.m.  ;  24°  50'  K.,  82°  18'  W. 
by  reckoning  from  preceding  noon  ; 

*  Qiauvenet's  "  Astronomy,"  I.,  214. 


174  NAVIGATION. 

T.  by  Watch  9h  30™  15s;  obs'd  altitude  of  0  58°  17'; 

Chro.  -  Watch  +  5*  12-  26s ;  Chro.  cor.  +  25™  155 ; 
Index  cor.  of  sextant  -f-  3'  20" ;  height  of  eye  18  feet ;  required 
the  longitude. 


h    m      s 

Q's  dec, 

Eq.  of  t. 

W.  T.   9  30  15 

o          -            II 

m     s 

C.-W.  5  12  26 

+  19  23  51.6  +  33/,.59 

+  3  48.81 

C.  C.    +  25  15 

r  100".8 

-.23 

G.  m.  t.,  May  17, 

+  1  45.2  - 

3  .4 

3  07  56  = 

:3U3           +  19  25  37 

I      1  .0 

+  3  48.6 

Eq.  of  t.           +3  48.6 

O         /       // 

/  // 

/  // 

G.  ap.  t.,  May  17, 

0     58  17           (I.e. 
+  14  30      (S.D. 

+  3  20 

Dip        —  4  09 

3  11  44.6 

+  15  51 

R.  &P.—     32 

h=    58  3130 

L  =    24  50 

1.  sec 

0.04214 

p=    70  34  23 

1.  cosec  0.02545 

2  s  =  153  55  53 

s=    76  57  57 

1.  cos 

9.35321 

s  —  h=    18  26  26 

1.  sin 

9.50012 

L.  ap.  t.,  May  16, 

8.92092 

21  45  45.3 

9  45  45.3 

1.  sin  %t  9.46046 

Long.           +  5  25  59.3 

or  81°  29'  50"  W. 

May  17,  noon,  lat.  by  mer.  alt.  of  O,  25°  8'  K ;  run  of  the 
ship  from  9 J*  a.m.  E.  N.  E.  (true)  18  miles. 

For  E.  N.  E.  18',  I  =  6'.9  N.,  p  =  16'.6  E.,  D  =  18'.4  E. 

At  the  time  of  the  a.m.  observations,  then,  the  latitude 
carried  back  from  noon  was  25°  1'  N.  Using  this  in  the  com- 
putation of  the  time,  we  find  the  L.  ap.  t.  May  16,  21*  45™  485.2, 
and  the  long.  81°  29'.1  W.  Applying  D  =  18'.4  E.,  we  have 
for  the  longitude, 

May  17,  noon,  81°  10'.7  W.,  from  observations  made  at 
9.45  a.m. 

By  p.m.  observations,  and  reduced  to  noon,  the  longitude 
was  found  to  be, 


LONGTTTJDE.  175 

May  17,  noon,  80°  44'  W.  from  observations  at  3.45  p.m. 

As  the  position  is  in  the  Gulf  Stream,  where  there  is  a 
strong  easterly  current,  the  difference  of  the  two  longitudes  is 
attributed  to  that  cause.  We  take,  then,  as  the  longitude  at 
noon, 

81°  10'.7  -  2-25*27'  =  81°  00'.5  W. 
o 

Note.  —  The  examples  under  Problem  45  can  be  adapted  to  this 
by  regarding  the  chronometer  correction  given,  instead  of  the  longitude. 

200.  3d  Method.  (Littrow's.  By  double  altitudes  of  the 
same  body.) 

When  two  altitudes  of  a  body  have  been  observed,  and  the 
times  noted  by  the  chronometer  or  watch,  the  hour-angles  and 
local  times  can  be  found  from  each  separately ;  and  thence  the 
longitude  for  each.  But  we  may  also  combine  them,  and  find 
the  hour-angle  for  the  middle  instant  between  them. 

Problem  47.  From  two  altitudes  of  a  heavenly  body, 
supposing  the  declination  to  be  the  same  for  both,  to  find 
the  mean  of  the  two  hour-angles,  the  latitude  ox  the  place 
and  the  Greenwich  time  being  given. 

Solution.  Take  the  mean  of  the  two  noted  times,  and  re- 
duce it  to  Greenwich  mean  time ;  and  find  for  it  the  declina- 
tion of  the  body. 

Eeduce  the  observed  altitudes  to  true  altitudes. 

Let  h  and  h'  be  the  two  altitudes, 

T  and  T',  the  corresponding  hour-angles  ; 

then  we  have,  by  (76), 

sin  h  =  sin  L  sin  d  -J-  cos  L  cos  d  cos  T, 
sin  N  **  sin  L  sin  d  -J-  cos  L  cos  d  cos  TT; 


176  NAVIGATION. 

and  by  subtracting  the  first  from  the  second, 

sin  N  —  sin  h  =  cos  L  cos  d  (cos  Tr  —  cos  T). 
By  Pl.  Trig.  (106)  and  (108),  this  reduces  to 

sin  J  (h!  —  h)  cos  i  (hr  -f  A)  = 

-cosicostfsin^r'  +  T)  sin  i  (T7'-:?7); 
whence 

2V  y  sin  J  (  Z"  —  T)  cos  L  cos  J         v      ' 

which  is  the  formula  used  in  Art.  300  (Bowd.). 

(T'  —  T)  for  the  sun  is  the  elapsed  apparent  time ;  for  a 
star,  the  elapsed  sidereal  time  ;  and  for  the  moon  or  a  planet, 
the  elapsed  sidereal  time  —  the  increase  of  right  ascension  in 
the  interval ;  and  can  be  found  from  the  difference  of  the  two 
chronometer  times. 

Then,  by  (131),  J  (Tr  +  T)  can  be  found,  and,  as  any  other 
local  hour-angle,  subtracted  from  the  corresponding  Greenwich 
hour-angle,  which  in  this  case  is  to  be  derived  from  the  mean 
of  the  noted  times. 

\  {Tr -\-  T)  is  +  or  —  according  as  the  second  altitude  is 
less  or  greater  than  the  first ;  so  that  it  is  on  the  same  side  of 
the  meridian  as  the  body  at  the  time  of  its  less  altitude. 

201.  The  method  presents  no  special  advantages  for  ob- 
servations on  shore,  except  in  the  case  of  two  nearly  equal 
altitudes  of  a  fixed  star  on  opposite  sides  of  the  meridian. 
In  the  case  of  the  sun  and  planets,  it  is  necessary  to  take  the 
change  of  declination  into  consideration  to  obtain  precise 
results. 

The  special  case  for  which  the  method  provides  is  at  sea, 
within  the  tropics,  when  the  sun  passes  the  meridian  at  a  high 
altitude.     In  that  case,  when  by  reason  of  clouds  observations 


LONGITUDE.  177 

near  noon  only  can  be  made,  or  it  is  desired  to  obtain  the 
longitude  as  near  noon  as  practicable,  let  a  pair  of  altitudes, 
or  several  pairs,  be  measured,  and  the  times  noted  with  all  the 
precision  practicable.  The  altitudes  should  be  reduced  to 
true  altitudes,  and  one  of  each  pair  for  the  run  of  the  ship  in 
the  interval  *  by  the  method  given  in  Prob.  53,  and  in  Bowd., 
Art.  288.  From  each  pair  the  middle  apparent  time  can  be 
found  by  (131),  and  the  mean  of  these  times  subtracted  from 
the  mean  of  the  Greenwich  apparent  times  for  the  longitude. 

202.  If  the  altitude  changes  uniformly  with  the  time,  or 
nearly  so,  the  mean  of  several  altitudes  observed  in  quick  suc- 
cession can  be  taken  for  a  single  altitude. 

If  the  observations  have  been  made  with  care,  the  errors 
of  instrument,  refraction,  and  dip  will  affect  the  two  altitudes 
of  each  pair  nearly  alike  ;  and  if  the  reduction  for  the  run  of 
the  ship  is  carefully  made,  the  difference  of  altitudes  in  com- 
parison with  the  difference  of  times  will  be  nearly  exact. 

203.  This  method  was  proposed  by  M.  Littrow,  Director 
of  the  Vienna  Observatory.  It  should  be  used  cautiously,  and 
the  errors  to  which  the  result  is  liable  in  any  case  carefully 
computed.  A  table  showing  the  error  of  time  which  may 
correspond  to  an  error  of  one  minute  in  each  of  the  observed 
altitudes,  when  t  =  30  inin.,  is  given  in  Art.  301  (Bowd.). 

Altitudes  greater  than  80°  and  an  interval  of  more  than 
half  an  hour  are  recommended,  but  an  intelligent  navigator 
can  readily  determine  when  he  can  safely  depart  from  these 
limits.  This  will  be  especially  the  case  when  the  altitudes 
are  on  both  sides  of  the  meridian. 

*  This  may  be  avoided,  if  the  course  of  the  ship  is  at  right  angles  to 
the  bearing  of  the  sun. 


178  NAVIGATION. 


Example. 

1.  1898,  May  16,  11.30  a.m.,  in  lat.  25°  15'  N.,  long.  56° 
20'  W.,  by  account ;  the  ship  running  N.  E.  (true)  8  knots  an 

hour. 

T.  by  Chro.    2h  32™  23s  O's  true  alt.,  81°    Y    0", 

"    "       "        2  53    11  "       "      "    83    40  30; 

Chronometer  correction  on  G.  mean  time  -|-  40w  51s ;  required 
the  longitude. 

The  distance  sailed  in  the  interval  is  2'.8.  The  sun's 
azimuth  at  the  1st  observation  is  found  to  be  N.  131°  E., 
which  differs  86°  from  the  course.  The  reduction  of  the  1st 
altitude  to  the  place  of  the  2d  is  (Prob.  53), 

2'.8  x  cos  86°  =  +  0'.2  =  +  12". 

h  m    s 
1st  chro.  t.  2  32  23  O's  dec.  Eq.  of  t. 

o     /    //  //  m  s  $ 

2d     "     "  2  5311        + 19  10 15.7  +34.4  +3  50.28—0.049 

Elapsed  chr.t. (r-T)=        2048  +157     (103.2  -17     (.15 

Mid.  "     "  2  42  47        +19  12  12.7/   13.8+3  50.1       (.02 


o 


Chro.  cor.  +  40  51 

G.  m.  t.  May  16  3  23  38  h  =81  01  12 

Eq.  t.  +  3  50.1  /i'=83  4030 

G.  ap.  t.  3  27  28.1  h  (h'-h)=  1 19  39      1.  sin  8.36489 

L.  ap.  t.  23  41  44  \  (h+h')=82  20  51      1.  cos  9.12439 

•.    .  (+3  45  44.1  i=25  15           1.  sec  0.04361 

Long.at2dobs.  |56  26  02W.  d=1912  13      1.  sec  0.02486 

T—  T  =     0  20  48    1.  cosec  £  t  1.34330 
I  (T'+T)  =  -01816    1.  sin  8.90105 

204.  4th  Method.  (By  equal  altitudes.)  Let  equal  altitudes 
of  a  heavenly  body  be  observed  east  and  west  of  the  meridian 
(Art.  175)  and  the  times  noted  as  in  other  observations ;  and 
the  mean  of  the  watch-times  in  each  set.  if  a  watch  is  used, 
reduced  to  chronometer  time.    If  both  sets  have  been  observed 


LONGITUDE.  179 

at  the  same  place,  and  the  declination  of  the  body  has  not 
changed,  the  mean  of  the  two  times  will  be  the  chronometer 
time  of  its  meridian  transit. 

If  the  declination  has  changed  in  the  interval,  as  is  ordi- 
narily the  case  with  the  sun,  moon,  or  a  planet,  the  correction 
for  such  change,  found  by  the  methods  of  Problem  46,  should 
be  applied. 

Applying  then  the  chronometer  correction,  we  have  the 
corresponding  Greenwich  time,  which  will  be  mean  or  sidereal 
as  the  time  to  which  the  chronometer  is  regulated. 

Finding  from  this,  by  the  method  in  Art.  189,  the  Green- 
wich hour-angle  of  the  body  (which  in  the  case  of  the  sun  is 
the  Greenwich  apparent  time),  we  have  the  longitude,  if  the 
first  observation  was  east  of  the  meridian,  as  the  correspond- 
ing local  hour-angle  is  then  0.  But  if  the  first  observation 
was  west  of  the  meridian,  the  local  hour-angle  is  12*,  and  must 
be  subtracted. 

This  method  should  be  used  on  shore,  when  practicable,  in 
preference  to  either  of  the  preceding. 

205.  Equal  altitudes  of  the  sun  can  be  conveniently  used 
at  sea  when  the  sun  passes  the  meridian  near  the  zenith ;  that 
is,  when  its  declination  and  the  latitude  are  nearly  the  same. 
Altitudes  very  near  noon  are  then  available  for  finding  the 
time  (Art.  196),  and  equal  altitudes  can  be  observed  with  only 
a  short  interval.  In  the  example  of  Art.  196,  an  interval  of 
eight  minutes  would  have  been  sufficient. 

If  the  ship  does  not  change  her  position  in  the  interval, 
the  middle  time  corresponds  to  apparent  noon ;  as  the  change 
of  declination  may  be  neglected,  unless  the  interval  between 
the  observations  is  so  great  as  to  require  it. 


180  NAVIGATION. 

206.  If  the  longitude  only  has  changed,  the  middle  time 
corresponds  to  apparent  noon  at  the  middle  meridian,  and 
will  give  the  longitude  of  that  meridian.  This  will  be  the 
longitude  at  noon,  if  the  speed  of  the  ship  has  been  uniform. 
But  if  it  has  not,  subtracting  half  the  change  of  longitude, 
when  the  true  course  is  west,  or  adding  it  when  the  course  is 
east,  will  give  the  longitude  of  the  place  where  the  first  alti- 
tude was  observed.  This  can  then  be  reduced  to  noon  by 
allowing  for  the  run  of  the  ship. 

If  the  change  of  longitude  is  west,  the  sun  arrives  at  the 
corresponding  altitude  of  the  afternoon  later  than  it  would 
do  if  observed  at  the  same  place  as  in  the  forenoon  ;  if  the 
change  is  east,  it  arrives  earlier;  and  the  difference  is  the 
time  of  the  sun's  passing  from  the  one  meridian  to  the  other ; 
that  is,  the  difference  of  longitude  expressed  in  time. 

If,  then,  2  t  is  the  elapsed  apparent  time, 

A  A,  the  change  of  longitude  (-J-  when  west), 
the  hour-angle  of  the  sun  at  each  observation  is  t  —  %  A\; 
and  (122)  becomes 

A  T  =  A„  dttsuiZ  A„  d  nan  d 

0  15sin(«-JAX)'+"l5tan(«-lAX)       {      } 

But  even  when  the  elapsed  time  is  so  great  that  it  is  thought 
necessary  to  correct  for  the  change  of  declination,  A  A  is  never 
large  enough  to  produce  a  change  of  Is. 

If  the  latitude  only  has  changed,  the  middle  time  requires 
correction  for  such  a  change,  which  can  be  deduced  in  a  simi- 
lar way  to  that  for  a  change  of  declination  in  Prob.  46.  But, 
as  in  the  fundamental  formula  (76), 

sin  h  =  sin  L  sin  d  -f-  cos  L  cos  d  cos  t, 


LONGITUDE.  181 

L  and  d  enter  with  the  same  functions,  the}  are  interchange- 
able.    If,  then, 
AA  L  is  the  hourly  change  of  latitude  (+  toward  the  north 

and  expressed  in  seconds),  and 
A'  T0 ,  the  required  correction, 

we  have  from  (122)  and  (124), 

=  _  ^Lttmd      AhL  t  tan  L 

0  15  sin  t      T      15  tan  t  K      } 

and  A'  T0  =  A  AhL  tan  d  +  B  A*  L  tan  X,  (134) 

for  which  Chauvenet's  tables  can  be  used. 

If  both  latitude  and  longitude  have  changed,  for  t  in  the 
denominators  of  (133),  we  may  substitute  t  —  J  A  A:  but  this 
at  sea  is  a  needless  refinement. 

The  restriction  of  this  method  to  a  short  interval  between 
the  observations  depends  upon  the  uncertainty  of  the  run  of 
the  ship,  and  consequent  imperfect  determination  of  Ah  X,  the 
mean  hourly  change  of  latitude  in  the  interval.  If  its  error 
is  supposed  to  be  -^  AA  X,  the  consequent  error  in  A'  T0  is  \  A'  T0. 

When  equal  altitudes  near  noon  are  practicable,  a  meridian 
altitude  of  the  sun  can  ordinarily  be  taken  for  latitude,  so 
that  L  will  be  sufficiently  exact.  Moreover,  the  latitude  and 
longitude  are  both  found  for  noon. 

Examples. 

1.  At  sea,  1898,  March  17,  noon,  lat.  by  mer.  alt.  of  the 
sun  3°  16'  S.,  long,  by  account  84°  58'  W. ;  equal  altitudes  of 
the  sun  were  observed  at  5h  34w  18s  and  6A  3m  24s  G.  mean 
time ;  the  ship  running  S.  S.  E.  (true)  10  knots  an  hour ;  re- 
quired the  longitude. 

For  S.  S.  E.,     10r,     A»  L  =  -  9r.2,     AA  A  =  -  3'.8. 


182  NAVIGATION. 


h    m    s 

O's  dec. 

Eq.  of  t. 

1st  G.  m.  t., 

5  34  18 

O       /        //              // 

m  s                  s 

2d  G.  m.t. 

6    3  24 

—  1  13  14  +59.29 

-  8  24.86  +  0.73 

Elapjed  t. 

0  29    6 

+       5  44  j  296.5 

+       4.23     (  3.65 
-  8  20.6      (    .58 

Mid.  G.  m.  t. 

5  48  51 

-  1  07  30  j  47.4 

—  Eq.  of  t. 

-  8  20.6 

AhL=—  552"      log 

2.742  n     log     2.742  n 

Mid.  G.  ap.  t. 

5  40  30.4 

L  =-3°16/ 

1.  tan  8.756  n 

Red.  for  A  L 

-I-  5.2 

d  =  —  1    07.5       1.  tan  8.293  n 

(  G.  ap.  t.  of 
\     or  Long. 

noon  5A  40" 

1  35s.6                           log  A  9.406  n     log  B  9.405 

85°  09' 

W.              j  -  2«.76 

0.441  n 

\  +  8  .00 

log     0.903 

In  this  example  the  sun's  azimuth  was  120°,  and  in  lm  the 

altitude  changed  13'.     An  inequality  of  30"  in  the  altitudes 

would  therefore  affect  the  result  only  ^  of  lm,  or  15.2.     An 

error  of  V  in  the  hourly  change  of  latitude  would  affect  the 

5s 
result  — ,  or  (K6. 

2.    At  sea,  1898,  June  29,  0^ ;  lat.  by  mer.  alt.  of  0, 33°  25' 
K,  long,  by  account,  147°  10'  E. ; 

near  11  a.m.,  T.  by  Chro.  1*  55m  548 


obs'dalt.  of  o  74°  9' 10"; 
1  p.m.,  "    "       "     3  45     0  )  u- 

Chro.  cor.  on  G.  m.  t.        -  36m  28s ;  In.  cor.  of  sex't  +  0'  50"  ; 

height  of  eye,  18  feet.     The  ship  run 

from  11  a.m.  to  noon  1ST.  3  p'ts  W.  11' ) 
from  noon  to  1   p.m.  N.  2     "    W.     8') 

required  the  longitude  at  noon. 


For  N.  3  W.  11' 
N.  2  W.    8 

whence 

A 

A 
A, 

L 
L 
L 

=  +  9'.1           A  X 

=  +  7  .4            A  a 
=  +  8  .25  =  495" 

=  +  TA 
=  +3.7 

A.  m.  Chro.  t.  4-  12A 
p.  if.  Chro.  t. 

h    m    s 
13  55  54 
15  45    0 

O's  dec. 

O        /        //              // 

+  23  16  50  -  7.3 

Eq.  oft. 
m    s            s 
-  2  59.8  —0.51 

Elapsed  time 
Mid.  Chro.  t. 
Chro.  cor.  (G.  m.  t.) 

149    6 
14  50  27 

-36  28 

-  1  44 
+  23  15  05 

-  7.2  (7.14 

-  307      |   .11 

Middle  long. 


LONGITUDE.  183 

h    m    s 
Mid.  G.  m.  t.,  June  28,  14  13  59  log  A&  L  2.695  log  AA  L  2.695 

—  Eq.  of  t.  —  3  07  log  A       9.410  n       log  B        9.398 

Mid.  G.  ap.  t.  14  10  52  1  tan  d     9.633  1.  tan  L    9.819 

Red.  for  A  L  +  27  log  1.738  n        log  1.912 

G.  ap.  t.  of  noon  14  11  19  -  54s.  7  4-  81s.  7  =  +  27s 

-  9  48  41 
or  147°  10'.2     E. 
Red.  to  noon  1  .8     W. 

Long,  at  noon  147     8  .4     E. 

The  sun's  azimuth  was  127° ;  for  A  t  =  lm,  A  h  =  10",  and 

an  inequality  of  1/  in  the  altitudes  will  affect  the  result  ?^  of 

27s 
lm,  or  3s.     An  error  of  V  in  Ah  L  will  affect  the  result  tt^z? 

or  3S.3. 

207.    5th  Method.    (By  transits.) 

Observe  the  transits  of  the  sun  or  a  star  across  the  threads 
of  a  well-adjusted  transit  instrument,  noting  the  times.  Re- 
duce the  mean  of  the  noted  times  for  semidiameter  and  errors 
of  the  instrument  as  in  Art.  184 ;  and  thence  find  the  Green- 
wich hour-angle  of  the  body  in  the  way  described  in  Art.  189. 
This  will  be  the  longitude,  if  the  upper  culmination  has  been 
observed,  as  the  local  hour-angle  is  0.  If  the  lower  culmina- 
tion has  been  observed,  the  local  hour-angle  is  12*. 

This  method  can  be  used  only  on  shore. 

Example. 

1898,  May  17,  17*  16m  20s.5  G.  mean  time,  the  meridian 
transit  of  a  Bootis  (Arcturus)  was  observed  ;  required  the 
longitude  of  the  place  of  observation. 


G.  mean  time  May  17 

17A  Wm  20s.  5 

80 

3   40    49.40 

Red.  for  G.  m.  t. 

+  2    50.24 

G.  sid.  t. 

21    00    00.14 

*'s  R.  A. 

14    11    03.73 

*'s  H.  angle  or  Long. 

+  6   48    56  .4     or  102°  14'  06' 

w. 


184  NAVIGATION. 

LONGITUDE.  — LUNAR    DISTANCES. 

208.  Problem  48.  To  find  the  longitude  by  the  distance 
of  the  moon  from  some  other  celestial  object. 

Solution.  If  we  have  given  the  local  mean  time  and  the 
true  distance  of  the  moon  from  some  celestial  object  as  seen 
from  the  centre  of  the  earth,  we  may  find,  by  interpolating 
the  Nautical  Almanac  lunar  distances  (Prob.  22),  the  Green- 
wich mean  time  corresponding  to  this  distance.  The  differ- 
ence of  this  from  the  local  time  is  the  longitude. 

The  local  time  may  be  found  for  the  instant  of  observation, 
either  from  an  altitude  of  a  celestial  object  observed  at  the 
same  time,  or  by  a  chronometer  regulated  to  the  local  time. 

At  sea  the  correction  of  the  chronometer  on  local  time  can 
be  found  from  altitudes  observed  near  the  time  of  measuring 
the  lunar  distance,  and  reduced  for  the  change  of  longitude 
in  the  interval  by  the  formula  (Art.  167), 

d  =  c  +  A  X, 

A  X  being  in  time  and  -j-  when  the  change  is  west. 

In  practice  the  apparent  distance  of  the  moon's  bright 
limb  from  the  sun  or  a  star  is  observed,  and  the  true  distance 
derived  by  calculation,  as  in  the  next  problem. 

209.  Problem  49.  Given  the  apparent  distance  of  the 
moon's  bright  limb  from  a  star,  the  centre  of  a  planet,  or 
the  sun's  nearest  limb,  to  find  the  true  distance  of  the 
moon's  centre  from  the  star,  or  the  centre  of  the  planet 
or  the  sun. 

Solution.  It  is  necessary  that  the  altitudes  of  the  two 
bodies  should  be  known,  either  directly  from  observations  at 
the  same  time,  or  from  observations  before  and  after,  and 


longitude: — lunar  distances.  185 

interpolated  to  the  time  of  observation  (Bowd.,  Art.  312) ; 
or  computed  from  the  local  time  (Prob.  32),  (Bowd.,  Art. 
313). 

The  Greenwich  time  is  also  supposed  to  be  known  approxi- 
mately, either  from  the  local  time  and  approximate  longitude, 
or,  as  is  preferable,  from  the  time  noted  by  a  Greenwich 
chronometer. 

A  complete  record  of  the  observations  will  include  the  ap- 
proximate latitude  and  longitude  of  the  place,  the  local  time 
and  chronometer  correction,  the  index  corrections  of  the  in- 
struments used,  the  height  of  the  barometer  and  thermome- 
ter, and  at  sea,  the  height  of  the  eye  above  the  water,  as  well 
as  the  noted  times  of  observation  and  the  observed  distances 
and  altitudes.  Several  observations  may  be  made  at  brief 
intervals,  and  the  means  taken. 

210.    The  preparation  of  the  data  embraces : 

1.  Finding  the  Greenwich  mean  time  approximately  from 
the  chronometer  time,  or  from  the  local  time. 

2.  Taking  from  the  Almanac  for  this  time  the  semi-diame- 
ter and  horizontal  parallax  of  the  moon,  and  of  the  other 
body  *  when  they  are  of  sensible  magnitude ;  adding  to  the 
moon's  semi-diameter  its  augmentation.     (Art.  59.) 

At  low  altitudes  the  contractions  produced  by  refractions 
should  be  subtracted  from  the  semi-diameters  of  the  sun  and 
moon.     Formulas  for  finding  these  are  given  in  Art.  213. 

When  the  spheroidal  form  of  the  earth  is  taken  into  con- 
sideration, to  the  moon's  equatorial  horizontal  parallax  (Art. 
57),  as  taken  from  the  Almanac,  should  be  added  the  augmen- 
tation to  reduce  to  the  latitude  of  the  place,  which  is  found 

*  The  sun's  horizontal  parallax  may  be  taken  as  8". 5. 


186  NAVIGATION. 

in  Table  19  (Bowd.).  The  decimations  of  the  two  bodies  to 
the  nearest  degree  are  required  from  the  Almanac  for  this 
purpose. 

3.  Applying  to  the  observed  distance  the  index  correction 
of  the  instrument,  and,  when  the  sun  is  used,  adding  the 
moon's  augmented  semi-diameter  and  the  sun's  semi-diame- 
ter ;  when  a  planet  or  star  is  used,  adding  the  moon's  aug- 
mented semi-diameter  if  its  nearest  limb  is  observed,  but 
subtracting  it  if  the  farthest  limb  is  observed. 

4.  Applying  to  the  observed  altitude  of  each  body  the 
index  correction,  dip,  and  semi-diameter  (when  necessary), 
so  as  to  find  the  apparent  altitude  of  its  centre.  If  the  true 
altitude  is  computed,  the  parallax  must  be  subtracted  and  the 
refraction  added. 

In  the  following  direct  method  it  is  necessary  also  to  find 
the  true  altitudes. 

211.    To  find  the  true  distance, 

let  D  =s  the  apparent  distance  of  the  centres, 

iy=  the  approximate  true  distance, 

h  =  the  apparent  altitude )    „ 

. ;■  .     ,  >of  3>'s  centre, 

h  =  the  true  altitude  J 

H=  the  apparent  altitude )    .  _  , 

,  \  of  O  s  centre,  planet,  or  star. 

H'=  the  true  altitude         ) 

In  Fig.  35,  let  m  and  S  be  the  apparent  places  of  the  moon 
and  other  body ;  mf  and  S',  their  true  places. 

The  true  and  apparent  places  of  each  are  on  the  same  ver- 
tical circle,  Z  m,  Z  S  respectively,  since  they  differ  only  by 
refraction  and  parallax,  which  act  only  in  vertical  circles, 
except  so  far  as  a  small  term  of  the  moon's  parallax  is  con- 
cerned, which  will  be  subsequently  considered. 


LONGITUDE.  —  L UNA R   DISTANCES. 


187 


==  90°  —  h    )  being  given, 


Fig.  35. 


Then    mS  =  D>    the    apparent 
distance ; 

mf  S'  =  17,  the  true  distance  ; 

and  in  the  triangle  m  Z  S, 

m$  =  D 

Zm 

Z  S  =  90°  -  H 

to  find  the  angle    Z,   we  have  by 
Sph.  Trig.  (32),* 

2         _  cos  \  (h  +  JJ+  />)  cos  }  (A  +  H-  J>) 
cos  h  cos  H 

Then  in  the  triangle  mf  Z  S', 

Zm'  =  90°  -  A'         and         ZS'  =  90°  -  J5P 
being  given,  mf  S'  may  be  found  by  Sph.  Trig.  (17),t 

sin2  i  J7  =  cos2  i  (//  +  H')  -  cos  A'  cos  H'  cos2  }  Z, 
or  by  substituting  the  value  of  cos2  \  Z,  and  putting 


s  =  i(h  +  ir+D), 


(135) 


sin2  J-  jy  =  cos2  J  (A'  +  H')  -  cos  A'  cos  H'  cos  «  cos  (s  -  D). 

cos  A  cos  If 

To  adapt  this  for  logarithmic  computation,  put 


then 


•   2  .  cos  N  cos  H'  ,         -r^ 

sir  \  m  = cos  s  cos  (s  —  Za), 

1  cos  h  cos  JI  y  J' 

sin2  £  2/  =  cos2  i  (*T  +  J5T )  -  sin2  £  m, 

which  by  Pl.  Trig.  (134),  becomes 

*  cos2  hA  =  sin  H<*  +  ft  +  c)  sin  £  (6  +  c  -  a)  # 

sin  6  sin  c 
t  sin2  ^  a  =  sin2  £  (6  +  c)  —  sin  6  sin  c  cos2  ^  A. 


(136) 


188 


NAVIGATION. 


sin2  i  jy  =  cos  i  (//  +  H'  +  m)  cos  i  (h!  +  H'  -  m), 

or,  if  we  put 

/=  i  (h'  +  H'  +  m),  (137) 

we  have 

sin  £  jy  =  V[cos  «'  cos  (/  —  m)].  (138) 

The  solution  is  effected  by  formulas  (135),  (136),  (137),  and 
(138). 

This  is  only  one  of  several  direct  trigonometric  solutions. 
It  is  easily  remembered,  involving  only  cosines  in  the  second 
members.  But  in  all  such  methods  7-place  logarithms  are 
required  for  the  computations. 

212.  If  the  moon's  augmented  parallax  has  been  used,  the 
distance  obtained,  Z^,  is  not  the  true 
distance  as  seen  from  the  centre  of  the 
'Z  earth,  but  from  the  point  C  (Fig.  36), 
where  the  vertical  line  of  the  place  in- 
tersects the  earth's  axis. 

A  reduction  to  the  centre,  C,  is  still 
required,  for  which  we  have  the  for- 
mula,* 


Fig.  36. 


A  Jy  =  A  P  sin  L 


(  sin  85        sin  8, 


■> 


(139) 


\sin  jy       tan  Dr 
in  which 

8S  is  the  sun's  declination, 
8TO,  the  moon's  declination, 
P,  the  moon's    equatorial  horizontal   parallax,  whose  mean 

value  is  57'  30", 
A,  a  coefficient  depending  on  the  eccentricity  of  the  terres- 
trial meridian,  the  mean  value  of  which,  for  latitude  45°,  is 
.0066855,  or  of  log  A,  7.8251, 

*  Chauvenet's  "Astronomy,"  I.,  399. 


LONGITUDE.  —  LUNAR   DISTANCES.  189 

A  sin  X,  the  distance  C  C,  with  CE  =  1. 

The  mean  values  of  A  P  =  23".07,  or  log  ^  P  =  1.3630 

may  be  used,  unless  great  precision  is  required. 

The  signs  of  the  declinations  and  latitude  are  +  when, 
north,  and  A  &  is  to  be  added  algebraically  to  D\ 

If  the  augmentation  of  the  parallax  has  been  neglected, 
the  distance  has  been  reduced  to  a  point  on  the  vertical  line 
between  C  and  C"  and  at  a  distance  from  A  equal  to  the 
equatorial  radius  C  E. 

213.  To  find  the  corrections  needed  for  the  contraction  by 
refraction  of  the  semi-diameters  of  the  sun  and  moon  in  the 
direction  in  which  the  distance  is  measured," 

let  q  =  the  angle  ZSm  (Fig.  35),  at  the  sun  or  star, 
Q  =  the  angle  Z  m  S,  at  the  moon, 
A  s  and  A's,  the  contractions  of  the  sun's  semi-diameter 

respectively  in  the  vertical  direction  S  Z,  and  in  the 

direction  of  the  distance  S  m ; 
A  aS  and  A'  S,  the  contractions  of  the  moon's  semi-diameter 

respectively  in  the  vertical  direction  m  Z,  and  in  the 

direction  of  the  distance  m  S. 

To  find  q  and  Q  from  the  three  sides  of  the  triangle  ZSm, 
putting,  as  in  (135), 

8  —  i  (h +  H+  D) 

We  haVG  *1A../  /coB,Bin(,-iry 


siM^y/f 


sin  |  q  = 


//cos  s  sin  (s  —  h)\ 
=  V  V    sin  D  cos  H   J 


(140) 


for  which  it  will  suffice  to  use  a  rough  approximation  of  D, 
and  for  the  computation,  logarithms  to  four  places ;  as  q  and 
Q  are  required  only  within  30r. 


190  NAVIGATION. 

The  contractions,  A  s  and  A  S,  of  the  vertical  semi-diameters, 
may  each  be  found  from  the  refraction  table,  by  taking  the 
difference  of  refractions  for  the  limb  and  centre. 

Then,  for  the  required  corrections,  we  have  the  formulas,* 

A's  =  As  cos2  q,  A'#  =  A  S  cos2  Q.  (141) 

This  contraction  for  either  body  is  less  than  1",  if  the  alti- 
tude is  greater  than  40°.  For  a  very  low  altitude,  it  is  best 
to  subtract  it  from  the  semi-diameter  in  the  preparation  of 
the  data,  so  that  D  will  be  corrected  for  it.  But,  unless  quite 
large,  it  will  suffice  to  compute  it  subsequently,  and  subtract 
it  from  D'  when  the  nearest  limb  is  used,  or  add  it  to  1/  when 
the  farthest  limb  is  used. 

214.  Let  A  D  =  the  reduction  of  the  apparent  distance  to 
the  true,  or  J7  =  D  +  A  D. 

A  great  variety  of  methods  have  been  given  for  finding 
A  D,  requiring  4-  or,  at  the  most,  5-place  logarithms ;  but  also 
needing  special  tables.  They  generally  neglect  to  take  into 
account  the  spheroidal  form  of  the  earth,  the  correction  of 
refraction  for  the  barometer  and  thermometer,  and  the  con- 
traction of  the  semi-diameters  of  the  sun  and  moon.  These 
together,  at  very  low  altitudes  and  in  extreme  cases,  may 
produce  an  error  of  3m  in  the  calculated  Greenwich  time, 
and  do  actually,  in  the  average  of  cases,  produce  errors  from 
10'  to  1*. 

In  1855,  Professor  Chauvenet  gave  a  new  form  to  the  prob- 
lem, with  convenient  tables,  by  which  all  these  corrections  are 
readily  introduced.  It  is  reprinted  in  a  pamphlet  with  his 
method  of  equal  altitudes,  and  it  is  also  given  in  Bowditch, 
Arts.  306  et  seq. 

*  Chauvenet's  "Astronomy,"  I.,  186. 


LONGITUDE.  —  LUNAR   DISTANCES.  191 

215.  The  moon's  mean  change  of  longitude  is  13°. 17640 
in  a  day,  or  33"  in  lm  of  time. 

An  error,  then,  of  33"  in  the  distance  will,  in  the  average, 
produce  an  error  of  lm  in  the  Greenwich  time,  or  15'  in  the 
longitude ;  or  an  error  of  10"  in  the  distance  will  produce 
an  error  of  about  20s  in  the  Greenwich  time,  or  5'  in  the 
longitude. 

We  may,  however,  readily  find  the  effect  of  an  error  of 
1",  and  thence  any  number  of  seconds,  in  the  distance,  by 
taking  the  number  corresponding  in  a  table  of  common  log- 
arithms to  the  "Prop.  Log.  of  Diff."  in  the  Almanac;  for 
this  prop.  log.  is  simply  the  logarithm  of  the  change  of  time 
in  seconds  for  a  change  of  1"  in  the  distance. 

216.  Errors  of  observation  are  diminished  by  making  a 
number  of  measurements  of  the  distance.  But  even  with 
a  skilful  observer  a  single  set  of  distances  is  liable  to  a  pos- 
sible error  of  10"  or  even  20". 

Errors  of  the  instrument  are  diminished  by  combining 
results  from  distances  of  different  magnitudes,  especially 
from  those  measured  on  opposite  sides  of  the  moon.  This 
cannot  usually  be  done  with  longitudes  at  sea,  but  may  be 
with  determinations  of  the  chronometer  correction.  The 
error  peculiar  to  the  observer,  that  is,  in  making  the  con- 
tacts always  too  close,  or  always  too  open,  is  not  eliminated 
in  this  way,  but  will  remain  as  a  constant  error  of  his 
results. 

The  accuracy  of  the  reductions  of  the  observed  to  the 
true  distance  depends  more  upon  the  precision  with  which 
the  differences  of  the  apparent  and  true  altitudes  —  that  is, 
the  parallax  and  refraction  —  have  been  introduced,  than 
upon  the  accuracy  of  the  altitudes  themselves. 


192  NAVIGATION. 

217.  Lunar  distances  are  rarely  used  at  the  present  day. 
They  are  given,  however,  in  the  Nautical  Almanac,  and  might 
possibly  be  used  for  rinding  the  Greenwich  mean  time,  with 
which  to  compare  the  chronometer.  They  may  thus  serve  as 
checks  upon  it,  which  in  protracted  voyages  might  be  much 
needed.  If  the  chronometer  correction  thus  determined 
agrees  with  that  derived  from  the  original  correction  and 
rate,  the  chronometer  has  run  well,  and  its  rate  is  confirmed ; 
if  otherwise,  more  or  less  doubt  is  thrown  upon  the  chro- 
nometer, according  to  the  degree  of  accuracy  of  the  lunar 
observation  itself.  If  the  discordance  is  not  more  than  20*, 
it  is  well  still  to  trust  the  chronometer,  as  the  best  observed 
single  set  of  distances  may  give  a  result  in  error  to  that 
extent.  If  it  is  large,  then  by  repeated  measurements  of 
lunar  distances,  differing  in  magnitude,  and  especially  on 
both  sides  of  the  moon,  and  carefully  reduced,  the  chro- 
nometer correction  can  be  found  quite  satisfactorily.  By 
taking  the  rate  into  consideration,  observations  running 
through  a  number  of  days  can  be  combined. 

218.  Other  lunar  methods  for  finding  the  longitude,  be- 
sides that  of  lunar  distances,  are, 

1.  By  moon  culminations,  or  observing  the  meridian  tran- 
sits of  the  moon  and  several  selected  stars  near  its  path, 
whose  right  ascensions  are  considered  well  determined. 

2.  By  occultations,  or  noting  the  instant  that  a  star  dis- 
appears by  being  eclipsed  by  the  moon,  or  that  it  reappears 
from  behind  the  moon.  The  first  is  called  an  immersion, 
the  second  an  emersion. 

3.  By  altitudes  of  the  moon  near  the  prime  vertical. 

4.  By  azimuths  of  the  moon  and  stars  observed  near  the 
meridian. 


LONGITUDE.  —  LUNAB  DISTANCES.  193 

These  methods,  except  occasionally  the  second,  are  avail- 
able  only  on  shore.  They  require  good  instruments,  careful 
observations  and  determinations  of  the  instrument  correc- 
tions, and  scrupulous  exactness  in  the  reductions,  especially 
those  which  involve  the  moon's  parallax. 

By  each  may  be  found  the  moon's  right  ascension,  and 
thence,  by  inverse  interpolation  in  the  Almanac,  the  corre- 
sponding Greenwich  mean  time.  Subtracting  from  it  the 
local  mean  time,  which  must  also  be  found  from  good  ob- 
servations, gives  the  longitude. 

If  corresponding  observations  are  made  at  two  different 
places,  their  difference  of  longitude  can  be  found  with  much 
less  dependence  on  the  accuracy  of  the  Ephemeris. 

When  the  two  local  times  of  the  occultation  of  the  same 
star  have  been  noted,  they  can  each  be  reduced  to  the  in- 
stant of  the  geocentric  conjunction  of  the  moon's  centre  and 
the  star  in  right  ascension ;  and  the  difference  of  the  reduced 
times  will  be  the  longitude. 

By  the  other  methods  the  change  of  the  right  ascension 
of  the  moon,  in  passing  from  one  meridian  to  the  other,  may 
be  found.  This,  divided  by  the  mean  change  in  a  unit  of 
time,  as  1*  or  lm,  computed  from  the  Ephemeris,  will  give  the 
difference  of  longitude  in  the  same  unit. 


194 


NAVIGATION. 


CHAPTER   IX. 

LATITUDE    AND   LONGITUDE    BY   SUMNER'S 

METHOD. 


Fig.  37. 


CIRCLES    OF    EQUAL    ALTITUDE.  -(  SUMN  EB'  S    METHOD.) 

219.  Suppose  that  at  a  given  in- 
stant the  sun,  or  any  other  heavenly 
body,  is  in  the  zenith  of  the  place  M 
(Fig.  37),  on  the  earth ;  and  let  A  A' A" 
be  a  small  circle  described  from  M  as 
a  pole.  The  zenith  distance  of  the 
body  will  be  the  same  at  all  places  on 
this  small  circle,  namely,  the  arc  M  A ; 
for  if  the  representation  is  transferred 

to  the  celestial  sphere,  or  projected  on  the  celestial  sphere 

from  the  centre  as  the  projecting  point, 

M  will  be  the  place  of  the  sun,  or  other  body,  and  the  circle 
A  A'  A"  will  pass  through  the  zeniths  of  all  places  on  the 
terrestrial  circle,  and 

M  A,  M  A',  etc.,  will  be  equal  zenith  distances. 

The  altitude  of  the  body  will  also  be  the  same  at  all  places 
on  the  terrestrial  circle  A  A'  A"  ;  hence  such  a  circle  is  called 
a  circle  of  equal  altitude. 

It  is  evident  that  this  circle  will  be  smaller  the  greater  the 
altitude  of  the  body. 


CIRCLES   OF  EQUAL   ALTITUDE.  195 

220.  The  latitude  of  M  is  equal  to  the  declination  of  the 
body,  and  its  longitude  is  the  Greenwich  hour-angle  of  the 
body ;  which,  in  the  case  of  the  sun,  is  the  Greenwich  appar- 
ent time,  or  24A  —  that  apparent  time,  according  as  the  time 
is  less  or  greater  than  12^.  This  is  evident  from  the  dia- 
gram, in  which,  regarded  as  on  the  celestial  sphere, 

P  M  is  the  celestial  meridian  of  the  place,  whose  zenith  is 
M,  and  its  co-latitude  ;  and  also  the  declination  circle,  and 
co-declination,  of  the  body  M ; 

and  if  P  G  is  the  celestial  meridian  of  Greenwich,  G  P  M  is, 
at  the  same  time,  the  longitude  of  the  place,  and  the  Green- 
wich hour-angle  of  the  body. 

If,  then,  the  Greenwich  time  is  known,  the  position  of  M 
may  be  found  and  marked  on  an  artificial  globe. 

221.  If,  moreover,  the  altitude  of  the  body  is  measured, 
and  a  small  circle  is  described  on  the  globe  about  M  as  a  pole, 
with  the  complement  of  the  altitude  as  the  polar  radius,  the 
position  of  the  observer  will  be  at  some  point  of  this  circle. 
His  position,  then,  is  just  as  well  determined  as  if  he  knew 
his  latitude  alone,  or  his  longitude  alone  ;  since  a  knowledge 
of  only  one  of  these  elements  simply  determines  his  position 
to  be  on  a  particular  circle,  without  fixing  upon  any  point  of 
that  circle. 

As,  however,  he  may  be  presumed  to -know  his  latitude 
and  longitude  approximately,  he  will  know  that  his  position 
is  within  a  limited  portion  of  this  circle.  Such  portion  only 
he  need  consider.     It  is  commonly  called  a  line  of  position. 

222.  The  direction  of  this  line  at  any  point  is  at  right 
angles  with  the  direction  of  the  body,  or  the  line  of  bearing,  as 
it  is  called  ;  for  the  polar  radius  M  A  is  perpendicular  to  the 


196  NAVIGATION. 

circle  A  A'  A"  at  A,  A',  A",  and  every  other  point  of  the 
circle. 

223.  Artificial  globes  are  constructed  on  so  small  a  scale 
that  the  projection  of  a  circle  of  equal  altitude  on  a  globe 
would  give  only  a  rough  determination.  But  the  projection 
of  a  limited  portion  may  be  made  upon  a  chart  by  finding  as 
many  points  of  the  curve  as  may  be  necessary,  and,  having 
plotted  them  upon  the  chart,  tracing  the  curve  through  them. 
The  portion  required  is  usually  so  limited  that,  when  the 
altitude  of  the  body  is  not  very  great,  it  may  be  regarded  as 
a  straight  line  ;  and  hence  two  points  suffice.  With  high  alti- 
tudes, three  points,  or  if  the  body  is  very  near  the  zenith,  four 
may  be  necessary,  and  even  the  entire  circle  may  be  required. 

224.  Problem  50.  From  an  altitude  of  a  heavenly  body 
to  find  the  line  of  position  of  the  observer,  the  Green- 
wich time  of  the  observation  being  known. 

Solution.  From  the  given  altitude,  and  assumed  latitudes 
L19  X2,  X8,  etc.,  differing  but  little  from  the  supposed  lati- 
tude, find  the  corresponding  local  times  (Prob.  37),  and  thence, 
by  the  Greenwich  time,  the  longitudes  \1}  A2,  A3,  etc.  Thus  we 
shall  have  the  several  points,  whose  positions  are  conveniently 
designated  as  (Zj,  A*,),  (X2,A2,),  (A>^3>)>  etc- 

It  facilitates  the  computation  to  assume  latitudes  differing 
10'  or  20',  as  the  £  sums  and  remainders  differ  5'  or  10',  and 
only  one  of  each  need  be  written. 

Or,  from  the  Greenwich  time  and  assumed  longitudes,  A1? 
A2,A3,etc,  find  the  corresponding  local  times  (Art.  77),  and 
thence  the  hour-angles  of  the  body  (Probs.  28,  29).  With 
these  and  the  observed  altitude,  find  the  corresponding  lati- 
tudes, ZjjZgjZ^etc.  (Prob.  40). 


CIRCLES  OF  EQUAL  ALTITUDE.  197 

This  is  more  convenient  than  the  preceding  method,  when 
the  body  is  near  the  meridian. 

In  either  mode  the  computation  may  be  arranged  so  that 
the  like  quantities  in  the  several  sets  shall  be  in  the  same 
line,  and  taken  out  at  the  same  opening  of  the  tables. 

The  several  points  may  then  be  plotted  on  a  chart,  each  by 
its  latitude  and  longitude,  and  a  line  traced  through  them, 
which  will  be  the  required  line  of  position.  Two  points  con- 
nected by  a  straight  line  are  sufficient,  unless  the  altitude  is 
very  great,  or  the  points  widely  distant. 

Thus  in  (Fig.  38),  let  A  and  B  be  two        Bfl    t,2 

such  points   plotted    respectively   on    the  P// 

parallels  of  latitude  L: ,  L2 ,  and  each  in  its  ~7~/ 

proper  longitude  ;  A  B  is  the  line  of  posi-  // 

tion.  and  the  place  of  observation   is  at        Att  _ 

rIG.  oo. 

some  point  of  A  B,  or  A  B  produced.  This 
is  all  which  can  be  determined  from  an  observed  altitude, 
unless  either  the  latitude,  or  the  longitude,  is  definitely  known. 
And  as  these  are  both  uncertain  at  sea,  except  at  the  time 
when  found  directly  by  observation,  the  position  of  the  ship 
found  from  a  single  altitude,  or  set  of  altitudes,  is  a  line,  of 
greater  or  less  extent  as  the  latitude,  or  the  longitude,  is  more 
or  less  accurately  known. 

In  uncertain  currents,  or  when  no  observations  have  been 
had  for  several  days,  the  extent  of  this  line  may  be  very  great. 
Yet,  if  it  is  parallel  to  the  coast,  it  assures  the  navigator  of 
his  distance  from  land ;  if  directed  toward  some  point  of  the 
coast,  it  gives  the  bearing  of  that  point. 

225.  If  there  is  uncertainty  in  the  altitude,  for  instance  of 
3',  the  line  of  position  having  been  computed  and  plotted, 
parallels  to  it  on  each  side  may  be  drawn  at  the  distance  of  3'. 


1 98  NA  VIGA  TION. 

So,  also,  if  there  is  uncertainty  in  the  Greenwich  time, 
parallels  may  be  drawn  at  a  distance  in  longitude  equal  to  the 
amount  of  uncertainty. 

In  either  case  the  position  of  the  ship  is  within  the  en- 
closed belt. 

In  Fig.  38,  a  b  is  such  a  parallel  to  the  line  of  position  A  B, 
its  perpendicular  distance  from  it  measuring  a  difference  of 
altitude ;  the  distance  A  a  on  a  parallel  of  latitude  measuring 
a  difference  of  longitude. 

226.  Since  the  line  of  position  is  at  right  angles  with 
the  direction  of  the  body  (Art.  222),  the  nearer  the  body  is 
to.  the  meridian  in  azimuth,  the  more  nearly  the  line  of  po- 
sition coincides  with  a  parallel  of  latitude ;  and  thus  a  posi- 
tion of  the  body  near  the  meridian  is  favorable  for  finding 
the  latitude  from  an  observed  altitude,  and  not  the  longi- 
tude. 

So  also,  the  nearer  the  body  is  to  the  prime  vertical,  the 
more  nearly  the  line  of  position  coincides  with  a  meridian, 
and  the  less  does  any  error  in  the  assumed  latitude  affect 
the  longitude  computed  from  an  observed  altitude.  So  that, 
if  the  body  is  on  the  prime  vertical,  a  very  large  error  in 
the  assumed  latitude  will  not  sensibly  affect  the  result.  Such 
a  position  of  the  body  is,  then,  the  most  favorable  for  find- 
ing the  longitude  from  an  observed  altitude. 

These  conclusions  have  been  previously  stated,  drawn 
from  analytical  considerations. 

227.  Two  or  more  points  of  a  line  of  position  as  (Lx ,  A.x), 
(X2 ,  A2),  etc.,  having  been  determined  by  Prob.  50,  if  the 
true  latitude,  Z,  be  subsequently  found,  the  corresponding 
longitude,  A,  may  be  obtained  by  interpolation. 


CIRCLES   01    EQUAL   ALTITUDE.  199 

Or,  the  place  of  the  ship  may  be  found  graphically  upon 
the  chart,  by  drawing  a  parallel  in  the  latitude,  X,  and  tak- 
ing its  intersection  P,  with  the  line  of  position  AB. 

So  also,  if  the  true  longitude,  A,  is 


subsequently  found,  the  corresponding 
latitude,  X,  may  be  obtained  by  interpo- 
lation ;  or,  a  meridian  E  F  may  be  drawn 
in  the  longitude,  A,  which  will  intersect 


F  B 


±2 


E    C 


the  line  of  position  in  P,  the  place  of        ■* 

.  .  Fig.  39. 

the  ship. 

If  there  is  uncertainty  in  either  of  these  elements,  two 
parallels  of  latitude  (as  in  Fig.  38),  or  two  meridians,  may 
be  drawn  at  a  distance  apart  equal  to  the  uncertainty. 

As  altitudes,  latitudes,  and  longitudes  are  never  found  at 
sea  with  much  precision,  and  may  under  unfavorable  circum- 
stances be  largely  in  error,  the  position  of  the  ship  on  the 
chart  is  not  properly  a  point,  but  a  belt,  more  or  less  limited 
according  to  the  accuracy  of  the  elements  from  which  it  has 
been  formed. 


228.    In  Fig.  39,  if  A  is  the  position  (Xx ,  Ax), 
B,  the  position       (X2 ,  A2), 
both  near  P,  the  true  position,  whose  latitude  is 
X,  and  longitude  is  A; 
we  have,  by  interpolation 

A2  —  Ai 

and 

as  the  formulas  for  finding  A,  the  longitude  of  the  true  posi- 
tion, when  its  latitude,  X,  is  known. 
Or,  we  have 


200  NAVIGATION. 

AX  =  AXX2~Xl 

A2  -  *!  (143) 

and  X  =  X,  -)-  A  Z 

as  the  formulas  for  finding  Z,  when  X  is  given.  The  several 
differences  are  most  conveniently  expressed  in  minutes  of 
arc,  or,  in  the  case  of  longitudes,  in  seconds  of  time.  The 
local  times  may  be  used  instead  of  the  longitudes  and  in- 
terpolated in  the  same  way. 

From  the  first  of  (142)  we  may  readily  determine  how 
much  a  supposed  error  in  an  assumed  latitude  affects  the 
resulting  local  time,  or  longitude. 


229.  Problem  51.  To  find  from  a  line  of  position  the 
azimuth  of  the  body  observed. 

Solution.  We  have  the  positions  (Zx ,  X{),  (Z2,A.2),  or  the 
latitudes  and  longitudes  of  two  points,  from  which  the  azi- 
muth, or  course  of  the  line  of  position,  can  be  found  by  middle 
latitude  sailing. 

Adding  or  subtracting  90°,  according  as  the  azimuth  of  the 
body  is  greater  or  less,  gives  the  azimuth  required. 

Or,  a  perpendicular  to  the  line  of  position  may  be  drawn 
upon  the  chart,  and  the  angle  which  it  makes  with  a  meridian 
may  be  measured  with  a  protractor.  The  azimuth  may  thus 
be  found  to  the  nearest  1°. 

230.  Problem  52.  To  find  the  position  of  the  observer 
from  two  altitudes  of  the  same  or  different  bodies,  the 
Greenwich  time  being;  known. 

Solution.  Find  the  line  of  position  from  each.  If  the  lines 
are  plotted  on  the  chart,  their  intersection  gives  the  position 
required. 


CIRCLES   OF  EQUAL   ALTITUDE.  201 

This  intersection  may  also  be  readily  found  by  computa- 
tion, when  the  lines  are  regarded  as  straight. 

Let 

(L\  \\)  (X'2  X.r2)  De  the  position  of  two  points  of  first  line, 
(L'\  k'\)  (Z"a  k"9)  "    "  «         "     "         "       "  second  line. 

A  L  and  A  A  be  the  run  in  lat.  and   long,  between  the  two 

observations 
(LX)  be  the  position  at  the  time  of  the  second  observation, 

the  upper  accents  distinguishing  the  observations,  the  lower 

accents  distinguishing  the  latitude  used  for  each  point. 

Then  by  Plane  Co-ordinate  Geometry,  assuming  (L\  X\)  as 
the  origin,  we  have, 

y-AZ  =  Z'*-^(x-AX)  (144) 

y  -  (L\  -  z\) = L"?rJ;>  (« -  ex",  -  vo)  (145) 


L  =  L\  +  y 
\  =  \\    +x 


}  ("«) 


(144)  is  the  equation  to  the  first  line,  moved  for  the  run. 

(145)  is  the  equation  to  the  second  line. 

(146)  is  the  intersection  of  the  first  line,  moved  for  the  run, 
with  the  second  line^  or  the  position  at  the  time  of  the 
second  observation. 

231.  Directions  N.  and  E.  are  to  be  marked  -f ,  and  those  S. 
and  W.  with  the  negative  sign.  If  both  lines  have  been  ob- 
tained by  simultaneous  observations  of  two  bodies,  A  L  and 
A  a  become  0,  and  if  the  same  assumed  latitudes  are  used  in 
both  observations,  of  course  L'\  —  L'  =  0. 

232.  The  more  nearly  perpendicular  the  lines  of  position 
are  to  each  other,  the  better  is  the  determination  of  their 


202  NA  VIGA  TION. 

intersection.  Hence,  the  nearer  the  difference  of  azimuths 
of  the  body  or  bodies  at  the  two  observations  is  to  90°,  the 
better  is  the  determination  of  position  from  double  alti- 
tudes. 

If  the  azimuths  are  the  same,  or  differ  180°,  the  two  lines 
of  position  coincide  in  direction,  and  there  is  no  intersection. 
In  this  case  the  great  circle  joining  the  two  bodies,  or  the 
two  positions  of  the  same  body,  is  an  azimuth  circle,  and 
passes  through  the  zenith.  An  approach  to  this  condition  is 
generally  to  be  avoided.  (Bowd.,  Art.  292,  note.)  Still, 
however,  if  the  two  bodies,  or  positions  of  the  same  body,  are 
near  the  meridian,  the  lines  of  position  nearly  coincide  with  a 
parallel  of  latitude.  The  latitude  is  then  well  determined, 
but  not  the  longitude.  If  the  two  bodies,  or  positions  of  the 
same  body,  are  near  the  prime  vertical,  the  lines  of  position 
more  nearly  coincide  with  a  meridian,  and  the  longitude  is 
well  determined  ;  but  not  the  latitude. 

When  the  difference  of  azimuths  is  small,  the  intersection 
of  the  two  lines  may  be  computed  with  tolerable  accuracy, 
while  it  cannot  be  definitely  found  by  the  projection  of  the 
lines  upon  a  chart. 

Examples. 

1.  1898,  Nov.  24,  about  6  a.m.  in  lat.  38°  40'  1ST.,  long. 
125°  W.  (approx.),  obs'd  alt.  Sirius  15°  27'  (West)  ;  chro.  t. 
2h  ±7m  43s.  Ran  thence  S.  58°.2  (true),  28.3  km  when  obs'd 
alt.  O  14°  15',  chro.  t.  5h  11 m  24s.  For  both  observations, 
the  chro.  cor.  is  —  25m  09s,  i.  c.  +  2'  30",  and  height  of  eye 
18  feet. 

Required  the  latitude  and  longitude  by  Sumner's  Method 
at  the  time  of  the  second  observation. 


CIRCLES   OF  EQUAL   ALTITUDE. 


203 


Solution. 


First  line  of  Position. 


Chro.  t. 
Chro.  cor. 


h   m    s 
2  47  43 


—  25  09 


*'sR.  A 


h  m     s  , 

6  40  43.6    ft  =15°  27' 


G.  m.  t.  Nov.  24  2  22  34 


^'sdec.   -16°  34' 30"  I.e.    +  2  30' 
h                                       h    m     s 

2.38    S0  16  13  51.54  Dip.   -  4  09 

Red.  G.  m.  t.     +  23  42  Ref.    -  3  28 

S'0  16  14  14.96  h  =  15  21  53 


1.  sec 
h=    15   21  53 
p  =  106  34  30     1.  cosec 
2  S  =  160   26  23 
S  =    80   13  12      1.  cos 
S  —  h=    64  51   19      1.  sin 

h  m     s 
>jc's  t.       =      3  35  22.5    1.  sin  |  t  9.65588 

$  R.  A.  =      6  40  43.6 

L.  s.  t.      =  10  16  06.1 

S'o  =  16  14  15 

L.  m.  t.  18  01  51.1  Nov.  23 

Long.  8  20  42.9  =  125°  10'.7  W. 


C    / 

0.10646  L\  =  38  50 

1.  sec  0.10848 

0.01843 

160  46  23 

0.01843 

9.23010 

80  23  12 

9.22271 

9.95676 

65  01  19 

9.95735 

9.31175 

9.30697 

9.65588 

3  34  05.5 
6  40  43.6 
10  14  49.1 
16  14  15 

18  00  34.1 

9.65348 

8  21  59.9  =  125°  30'  W. 


/Second  line  of  Position. 

Ean  S.  58°,  E.  28r.3 ;    A  lat.  15  S. ;  dep.  =  24  E. ;  A  long. 
=  31  E.     Lat.  left,  38°  40r  N. ;  lat.  in,  38°  25'  N. 

Chro.  t.                 5*  11™  24s  Q       14°  15'             I.  c.  +  2'  30" 

Chro.  cor.              —  25    09  +  10 .59        S.  D.     16  15 

G.  m.t.,Nov.  24,  4   46    15    =  4*.77  h  =  14   25  .59        Par.            09 

Eq.  t.                      +  13    02 .1  Dip    —  4  09 

G.  ap.  t.                 4   59    17.1  Ref.   —  3  46 


Q'sdec.  —  20°37'39".8 
—  2  23  .2 
-  20   40  03 


•  30.02  Eq.  t.   +  13™  05s. 67  -  0s. 739 

120.1  (2 .96 

21.0  ~3-53  J     .52 

2.1  +  13    02.14  I     .05 


204 


4 

NAVIGATION. 

£"i 

38°  15' 

1.  sec     0.10496 

L"2  =  38°  35' 

0.10696 

ft 

14   25  59'' 

p 

110   40  03 

1.  cosec  0.02889 

0.02889 

2s 

163   21  02 

163°  41'  02" 

s 

81   40  31 

1.  cos      9.16072 

81   50  31 

9.15201 

-h 

67   14  32 

1.  sin      9.96481 
9.25938 

67    24  32 

9.96533 
9.25319 

ap.  t. 

20*  38w  09s 

d  sin  it  9.62969 

20*  39™  40s.7 

9.62660 

Long.       8   21    08.1=  125°  17' W. 


8A  19™  36.4  =  124°  54'.  1  W. 


To  compute  the  lat.  and  long,  of  the  intersection  of  the  first 
line  moved  for  the  run,  with  the  second  line. 

L\  =  38°  30'N.  L'2  -L\  =  +20'  l\  =  125°  10/7  W.  V2  -l\    =  -19'.3 
L'2  =  38   50        L"2  -L\  =  +20  2'9  =  125   30  l"2  -l'\  =  +22  .9 

L'\  =  38   15         L'\  -L\  =  -15  X'\  =  125   17  l'\  -l\  =  -  6  .3 

Z"2  =  38  35  AX  =-15  *"2=124  54.1  A*  =+31 


Then  by  substitution  in  (144)  and  (145)  we  have 

V  +  15  = 
from  which 


20_  (x  _  31)  and  y  +  15  =  —  (x  +  6.3) 


19.3  v  7  "  22.9 

x  =  +  13.9  and  y  =  +  2.7. 

Therefore  the  position  at  the  time  of  the  second  observation 
is  by  (146) 

L  =  38°  30'  +  2'.7  N.  =  38°  32'.7  N. 

\  =  125°  10'.7  W.  +  13'.9  E.  =  124°  56'.8  W. 

2.  1898,  Aug.  19,  at  sea,  making  passage  from  Honolulu  to 
San  Francisco:  position  at  noon,  lat.  33°  15' N.,  long.  135° 
40'  W.  Thence  ran  N.  62°  E.  (true)  202  kn.  until  about  8  a.m. 
Aug.  20,  when  obs'd  alt.  O  37°  01',  chro.  t.,  5h  37™  375 ;  chro. 
cor.  -  18™  175 ;  i.  c.  -  2'~30"  ;  height  eye,  25  ft. 

Ran  thence  N.  62°  E.  (true)  28  kn.  until  about  10.45  a.m., 
when  obs'd  alt.  O  64°  25',  chro.  t.  lh  16m  40s ;  c.  c,  i.  c,  and 
dip  as  before. 


CIRCLES   OF  EQUAL  ALTITUDE. 


205 


Made  noon,  Aug.  20,  after  running  9  kn.  farther  on  the  same 
course.     No  meridian  observation. 

Kequired  the  position  at  noon,  Aug.  20,  and  the  set  and 
drift  of  the  current. 


SOLUTION. 


N. 

Noon  — 8  a.m.     N.  62°  E.      202' 

Noon,  Aug.  19,  Lat.  33°  15'     N. 

Al    1    34  .8  N. 

8  a.m.,  Aug.  20,         Lat.  34   49  .8  N. 


E 

A  X  =  215'.3 
Long.  135°  40'     W. 
A  A.         3   35.3  E. 
Long.  132   04  .7  W.  by  D.  R. 


First  line  of  position. 

Chro.  t.     5*  37m  37s 

Q.     31 

°01' 

fS.D.  +  15' 51" 

C.  c.           -  18    17 

+  7  17" 

Dip      —  4  54 

G.  m.  t.      5    19    20 

=  5A.32,       h  =  Sht 

'   08  17   J 

Par.              07 

Eq.  t.          —    3    08  .5 

Aug.  20 

Ref .     -  1  17 

G.  ap.  t.      5   16    11  .5 

[i.  c.     -  2  30 

©'s  dec.  +  12°  22' 

01".7       -  49".64 

Eq.  t  —  3W  lls.63     +  0*. 59 

r248".2 

r2.95 

-4 

24  -1      \    14  .9 

+  3.14            >18 

+  12   17 

37  .6      I     1 

-  3    08  .5       I     .01 

L\  =    34°  30' 

L  sec      0.08401 

L'2  =35°           1.  sec  0.08664 

h  =    37   08  17" 

p  =    77   42  22 

1.  cosec  0.01008 

0.01008 

2  s  =  149   20  39 

149°  50'  39" 

s  =    74   40  20 

1.  cos      9.42217 

74   55  20         9.41520 

s-  h  =    37   32  03 

1.  sin      9.78479 

37  47  03         9.78724 

9.30105 

9.29916 

1.  ap.  t.      20*  27m  28s.3 

1.  sin  It  9.65053 

20*  27™  58s. 3      9.64958 

G.ap.t.       5   16    11.5 

5    16    11.5 

8   48    43.2 

8   48    13.2 

A'i  =  132°10'.8) 
L\=    34   30     j 

A/2  =132°03'.3  / 

L'2=  35 

00      j 

Second  line  of  Position. 

Ean  8  —  10.45  a.m.  N.  62°  E.  28';    A  L  =  13.1  Dep.  = 
24.7,  A  a  =  30'.1  E. 


206 


NAVIGATION. 


Mean  long,  from  1st  line  132°  07'  W.  at  8  a.m. 

Approx.  long,  at  10.45  a.m.  131°  37  W.  =  Sh  46™  28s. 

By  D.  R.  at  10.45  a.m.  lat.  35°  02'.9  N.,  long.  131°  34'.6  W. 


Chro.  t.  8  16  40        Q   64  25  00 

f  S.  D.  +  15  51   Dip 

-4  54 

C.  c.   -  18  17             +8  03 

\  Par.  + 

4   Ref 

.  -  28 

G.  m.  t.  7  58  23  =  7^.97,  h  =  64  33  03 

I 

I.e. 

-2  30 

Eq.  t.   —  3  06.9  Aug.  20 

G.ap.t.  7  55  16.1          Q's  dec. 

Eq. 

of  t. 

Mean  \  8  46  28      +  12°  22'  01".7  - 

.  49". 64   -  3" 

n  11s.  63 

+  0*.  59 

Meant.  0  51  11.9                j 

347".5 

c   4s.  13 

-  6  35  .7  \ 

44  .7 

+  4.70 

|   .53 

+  12  15  26   t 

3  .5    -  3 

06.9 

I  .04 

h'     =    64°  33'  03"               1. 

sin   9.95567 

tx     =    12  33  00    1.  sec  0.01050 

d      =    12  15  26    1.  tan  9.33696    1. 

cosec  0.67305 

<£"  =  12  32  51    1.  tan  9.34746    1. 

sin   9.33695 

<£'  =  22  29                  1. 

cos.  9.96567 

l/\-    35°01'.9) 

X"i  =  131  22  )  t2    =   13°  03'  00"   1. 

1.  sin 

9.95567 

sec  0.01136 

1. 

tan  9.33696 

1.  cosec 

i  0.67305 

<£"  =  12  34  18    1. 

tan  9.34832 

1.  sin 

9.33778 

$    =   22  13 

1.  cos 

9.96650 

L"2=   34°47,.3) 

A"*  =131  52   J 

To  compute  the  position  at  10.45  A.  m. 


L'\  =34°  30'  N. 
L'2  =35  00  N. 
L"\  =35  01  .9  N. 
L"2   =34  47  .3  N. 

A  L  =    +  13M 


A'i  =  132°  10'.8  W. 
A'a  =  132  03  .3  W. 
A"i  =  131  22  W. 
A"2  =  131  52  W. 
AA=   +30M 


L'2  -L'\  = 
L"2  -  L"\  = 
L"\  -L'x   = 


30' 

14.6 

31.9 


A'2  —  A'i  =  +  7'.5 
A"2  — A"i  =  —  30 
A"!  -  a'x  =  +  48  .8 


Then  by  substitution  in  (144)  and  (145) 


y  -  13.1  = 
y  -  31.9  - 


-f30 


(x  -  30.1)  =  4  (x  -  30.1) 


+  7.5 
-  30    {X      ^'*}       15 


whence 


x  =  +  32.9  and  y  =  +  24.3. 


CIRCLES   OF  EQUAL  ALTITUDE. 


207 


by  (146)  L  =    34°  54.3  K 

X  =131°  37.9  W. 


at  10.45  a.m. 


10.45  a.m.  lat.  35°  02.9  N.         long.  131°  34.6  W.  by  D.  R. 
"      "    34  54.3  N.  M      131  37.9  "     "  obs. 

A  L  8.6  S.        A  A  3.3  W.  dep.  =  2.7 

Current  for  22f  hrs.  9  kn.  S.  17°  W.  =  .4  kn.  per  hour. 

N.         S.        E.        W. 


10.45  —  noon         N.  62°  E.      9. 
Current  to  noon      S.  17   W.     0.5 

10.45  a.m.  lat.        34°  54.3  H". 
Run  to  noon  A  L  3.7  N. 

Noon  Aug.  20  lat.  34  58     N. 


4.2  7.9 

_    as    _    02 

3.7  N.        7.7  AX  =  9.4E. 

long.  131°  37.9  W.  by  obs. 

A  X  9.4  E. 

long.  131   28.5  W. 


233.    The  correction  for  the  run  of  the  ship  between  two 
observations  may  be  determined  as  follows.    (Art.  288,  Bowd.) 

Problem.    53.     To  reduce  an  observed  altitude  for  a 
change  of  position  of  the  observer. 

Solution.     Let 

Z  (Fig.  40)  be  the  zenith  of  the  place  of  observation ; 
h  =  90°  —  Zm,  the  observed  altitude ; 
Z',  the  zenith  of  the  new  position ; 

h'  =  90°  —  7/  m,  the  altitude  reduced  to  the  new  position,  Zr. 
d  =  ZZ',  the  distance  of  the  two 

places,   here    referred    to 

the  celestial  sphere ; 

C=  P  Z  Z',  the  course; 
Z=PZ»i,  the  azimuth  of  m ; 
Z  —  0=  m  Z  Z',  the  difference  of  the 
course  and  azimuth. 

Z  Z',  being  small,  may  be  regarded  as 

a  right  line, 
Z  Z'  O  as  a  plane  right  triangle,  and 


208  NAVIGATION. 

0  m,  without  material  error,  as  equal  to  71  m ;  so  that  we 

shall  have 

ZO  =  ZZ'cosZ'Zm 

Z'wi=Zm-ZO 

or  putting  A  h  =  Z  O, 

Ah=dcos(C-Z)   X 
h'  =  h+Ah  I  {    V 

A  A  =  d  cos  (  (7  —  Z)  is,  then,  the  reduction  of  the  observed 
altitude  to  the  new  position  of  the  observer:  it  is  additive 
when  C  —  Z  <  90°  numerically ;  subtractive  when  C  —  Z> 
90°.  It  is  smaller,  and  can,  therefore,  be  more  accurately 
computed  the  nearer  C  —  Z  approaches  90°.  It  is,  therefore, 
better  to  reduce  that  altitude  for  which  the  difference  of  the 
course  and  azimuth  is  nearest  90°. 

If  the  second  is  the  one  reduced,  then  (7  is  the  opposite  of 
the  course. 

In  practice  Z  Z'  does  not  usually  exceed  30',  so  that  al- 
though an  arc  of  a  great  circle  of  the  celestial  sphere,  it  may 
be  regarded  as  representing  the  distance,  d,  of  the  two  places 
on  the  earth ;  or,  at  sea,  the  distance  run.  The  azimuth,  or 
bearing,  of  the  body  can  be  observed  with  a  compass,  or  be  com- 
puted to  the  nearest  degree,  or  half-degree,  from  the  altitude. 

The  assumption,  7J  m  =  O  m,  is  more  nearly  correct,  the 
greater  Z'morZ  m,  that  is,  the  smaller  the  altitude.  If  we 
treat  ZZ'wasa  spherical  triangle,  d  =  Z  Z'  being  expressed 
in  minutes  and  still  very  small,  we  shall  find 

A  h  =  d  cos  (O  -  Z)  -  \  d2  sin  V  tan  h  sin2  (O  -  Z)  ;   (148) 

but  the  last  term  is  inconsiderable  unless  d  and  h  are  both 
large.  For  instance,  if  d  =  30',  it  will  not  exceed  V  unless 
h  >  82°. 


CIRCLES   OF  EQUAL  ALTITUDE.  209 

Example. 

The  two  altitudes  of  the  sun  are  36°  16'  20",  586  15'  20", 
the  compass  bearings  of  the  sun  respectively  S.  E.  by  E.  £  E. 
and  W.  S.  W. ;  the  ship's  compass  course,  and  distance  made 
good  in  the  interval  N.  K  W.  £  W.  25  miles ; 

S.  &i  E.  differs  from  N.  2\  W.  13  points,  so  that  the  re- 
duction of  the  1st  altitude  to  the  position  of  the  2d  is 

25'  x  cos  13  pts.  =  -  25'  cos  3  pts.  =  -  20'.8  =  -  20'  48". 

S.  6  W.  differs  from  S.  2\  E.  8£  points,  and  the  reduction 
of  the  2d  altitude  to  the  position  of  the  1st  is 

25'  cos  8£  pts.  =  -  25'  cos  1\  pts.  =  -  2'  30" 

or  —  2'  39",  if  the  last  term  of  (148)  is  included. 

234.  By  (147)  or  (148)  we  may  reduce  one  of  the  two 
altitudes  for  the  change  of  the  ship's  position  in  the  interval. 
But  instead  of  this  we  may  put  down  the  line  of  position  for 
each  observation,  and  afterwards  move  one  of  them  to  a  par- 
allel position  determined  by  the  course  and  distance  sailed  in 
the  interval.     Thus  in  Fig.  41,  let 

A  B  be  the  line  of  position  for  the  first 
observation,  and 

Aft  represent  in  direction  and  length 
the  course  and  distance  sailed  in 
the  interval ;  then 
a  b,  drawn  parallel  to  A  B,  is  the  line  of  position  which 
would  have  been  found  had  the  first  altitude  been  observed 
at  the  place  of  the  second. 

If  the  second  observation  is  to  be  reduced  to  the  place  of  the 

first,  then  A  a  in  direction  must  be  the  opposite  of  the  course. 

The  perpendicular  distance  of  AB  and  a  b  is  the  reduc- 


210  NAVIGATION. 

tion  of  the  altitude  for  the  change  of  position :  for  that  dis- 
tance is  A  a  X  cos  (B  A  a  —  90°). 

235.  There  are  several  other  methods  of  finding  the  lati- 
tude by  two  altitudes  either  of  the  same  or  of  different 
bodies ;  but,  with  the  exception  of  the  one  following,  their 
methods  are  so  intricate  and  Sumner's  Method  has  proved  so 
valuable  a  substitute  for  them,  that  they  are  rarely  if  ever 
used  at  the  present  time.  Four  methods  are  given  in  Bow- 
ditch,  Arts.  288  to  292  ;  a  full  discussion  of  the  principles 
upon  which  they  are  based  and  of  a  method  by  three  alti- 
tudes may  be  found  in  Chauvenet's  Astronomy. 

236.  Problem  54.  To  find  the  latitude  by  the  rate  of 
change  of  altitude  near  the  prime  vertical  (PresteVs  Meihoa)* 

In  the  note  to  Art.  197  we  have,  for  a  very  brief  interval  of 
time,  and  a  small  change  of  altitude, 


or,  T'-T=t 


15  cos  L  sin  Z 
h'-h 


15  cos  L  sin  Z 


whence  cos  L  = cosec  Z;  (149) 

±o  t 

in  which  H  —  h  is  expressed  in  seconds  of  arc  and  t  in  seconds 

of  time,  and,  Z  being  -f-  when  east,  —  when  west,  cos  L  is  always 

positive.     If  Z  is  near  90°,  its  cosecant  varies  slowly.     When 

Z  =  90°  we  have,  , ,      h 

COs.L=  — — .  (150) 

If,  then,  two  altitudes  are  carefully  observed  near  the  prime 
vertical,  and  the  times  noted  with  great  precision,  the  interval 

*  Chauvenet's  Astronomy,  I.,  303,  311. 


CIRCLES   OF  EQUAL   ALTITUDE.  211 

not  exceeding  8  or  10  minutes,  an  approximate  latitude  may  be 
found  by  (150),  when  the  altitudes  are  within  2°  or  3°  of  the 
prime  vertical ;  or  by  (149)  when  they  are  at  a  greater  distance, 
and  Z  is  approximately  known. 

The  time  of  passing  the  prime  vertical  can  be  found  by  (86). 
Z  may  be  roughly  computed  from  the  altitudes,  or  found  within 
2°  from  the  bearing  observed  by  a  compass,  which  will  suffice, 
if  the  observations  are  made  within  10°  of  the  prime  vertical. 

As,  near  the  prime  vertical,  the  altitude  changes  uniformly 
with  the  time,  several  altitudes  may  be  observed  in  quick  suc- 
cession, and  the  mean  taken  as  a  single  altitude. 

The  larger  h!  —  h  and  t,  consistent  with  the  supposition  of 
uniformity  of  change  and  the  condition  by  which  they  are  sub- 
stituted for  their  trigonometric  functions,  the  more  accurate  in 
general  will  be  the  result. 

Still  the  method  does  not  admit  of  much  precision.  It  is 
entirely  unavailable  near  the  equator,  and  in  latitude  45°  may 
give  a  result  in  error  from  5  to  10  minutes,  even  when  the 
greatest  care  has  been  bestowed  on  the  observations.  It  may, 
however,  be  useful  to  the  navigator  in  high  latitudes,  as  it  can 
be  used  for  altitudes  of  the  sun,  when  almost  exactly  east  or 
west,  and  it  will  restrict  the  position  of  the  ship  to  a  limited 
portion  of  the  line  of  position  found  by  Sumner's  Method. 
There  are  occasions  at  sea,  when  to  find  the  latitude  only  with- 
in 10'  is  very  desirable. 

Examples. 
1.  1898,  June  15,  7h  a.  m.,  in  lat.  60°  N.,  long.  60°  W. ; 

T.  by  Chro.  11*  13™  25*.3,     obs'd  alt.  O  27°  00'  23")  O's  Az. 
"    "       "      11    19    51.0,        "       "      "   27   48  42   j  N.  88°  E.; 

required  the  latitude. 


212  NAVIGATION. 

tV  log         8.8239 

h'-h  =  48'  19"  log  3.4622 

t  =  6m  25s.  7  ar.  co.  log  7.4137 

Z  -  88°  1.  cosec  0.0003 

L  =  59°  55'  N.  1.  cos     9.7001 

If  A  (H  -  h)  =  10",  A  log  (N  -  h)  =  A  1.  cos  L  =  .0015, 
and  A  X  =  6'.  If  the  difference  of  altitudes  can  be  depended 
on  within  5",  the  latitude  is  correct  within  3'. 

2.  1898,  July  13,  5h  p.m.,  in  lat.  54°  20'  N.,  long.  113°  W., 
by  account ;  the  altitude  of  the  sun's  lower  limb  was  observed 
at  0*  23m  34*  by  the  chronometer,  which  was  slow  of  G.  mean 
time  10w  18s ;  and  the  sextant  remaining  clamped,  the  upper 
limb  arrived  at  the  same  altitude  at  0h  27m  88.5 ;  the  true  alti- 
tude of  both  limbs  was  27°  18'  20"  ;  required  the  latitude. 

The  sun's  diameter,  31' 33",  is  the  difference  of  altitudes 
in  this  case.  The  sun's  azimuth  computed  from  the  altitude 
and  supposed  latitude  is  N.  88 %°  W. 


tV 

log         8.8239 

ti  =  31'  33" 

log          3.2772 

t  =  3m  34*.5 

ar.  co.  log  7.6686 

Z  =  88£° 

1.  cosec  0.0002 

L  =  53°  56'  N. 

1.  cos     9.7699 

If  we  suppose  t  to  be  in  error  l8,  1.  cos  L  will  be  in  error 
.0020  and  L  11'.  If  the  elapsed  time  can  be  depended  on 
within  0*.5,  the  latitude  is  correct  within  6'. 

The  longitude  obtained  from  the  same  observations  is 
113°  5r  W. 

This  method  of  observing  the  successive  contacts  of  the 
two  limbs  of  the  sum  with  the  horizon  with  the  sextant 
clamped  is  recommended. 


CIRCLES   OF  EQUAL  ALTITUDE.  213 

CHAPTER   X. 

AZIMUTH   OF  A  TERRESTRIAL   OBJECT. 

237.  In  conducting  a  trigonometric  survey,  it  is  necessary 
to  find  the  azimuth,  or  true  bearing,  of  one  or  more  of  its 
lines,  or  of  one  station  from  another.  Thence,  by  means  of 
the  measured  horizontal  angles,  the  azimuths  of  other  lines 
or  stations  can  be  found ;  and,  still  further,  a  meridian  line 
can  be  marked  out  upon  the  ground,  or  drawn  upon  the 
chart. 

For  example,  suppose  at  a  station,  A,  the  angles  reckoned 
to  the  right  are 

B  to  C,  48°  15'  35" ;   CtoD,  73°  37'  16" ;  D  to  E,  59°  45'  20" ; 

and  that  the  azimuth  of  D  is  N.  35°  16'  15"  E. ;  the  azimuths 
of  the  several  lines  are 

A  B,  K  86°  36'  36"  W.        A  B,  K  35°  16'  15"  E. 

A  O,  K  38    21     1W.        A  E,  K  95      1  35   E. 

If  upon  the  chart  a  line  be  drawn,  making  with  A  B  an 
angle  of  86°  36'  36"  to  the  right,  or  with  A  B  an  angle  of 
35°  16'  15"  to  the  left,  it  will  be  a  meridian  line. 

Or,  if  a  theodolite  or  compass  be  placed  at  A  in  the  field, 
and  its  line  of  sight,  through  the  telescope  or  sight-vanes,  be 
directed  to  B,  and  the  readings  noted,  and  then  the  line  of 
sight  be  revolved  to  the  left  until  the  readings  differ  35°  16' 


214 


NAVIGATION. 


15"  from  those  noted,  it  will  be  directed  north.  If  a  stake 
or  mark  be  placed  in  that  direction,  it  will  be  a  meridian 
mark  north  from  A. 


238.  If  the  azimuth  of  a  terrestrial  object  is  known,  it 
may  be  conveniently  used  in  finding  the  magnetic  declina- 
tion, or  variation  of  the  compass.  For,  let  the  bearing  of  the 
object  be  observed  with  the  compass  ashore  —  the  difference  of 
this  magnetic  bearing  and  the  true  bearing  is  the  magnetic 
declination,  or  variation,  required.  It  is  east  if  the  true  bear- 
ing is  to  the  right  of  the  magnetic  bearing  ;  but  west  if  the 
true  bearing  is  to  the  left  of  the  magnetic  bearing.* 


239.  Problem  55.  To  find  the  azimuth,  or  true  bear- 
ing of  a  terrestrial  object. 

Solution.     Let 

Z  (Fig.  42)  be  the  zenith,  or  place,  of 

the  observer; 
O,  the  terrestrial  object ; 
M,  the    apparent   place  of  the  sun,  or 

some  other  celestial  body ; 
Z  =  N  Z  0,  the  azimuth  of  0  ; 
z  =  1ST  Z  M,  the  azimuth  of  M ; 
f  =  Z-2=MZ0,  the  azimuth  angle 

between  the  two  objects,  or  the  difference  of  azimuth  of 

M  and  0. 
The  problem  requires  that  z  and  £  be  found ;  then  we  have 

*  This  has  reference  to  the  two  readings.  The  actual  direction  of 
the  object  is  the  same;  hut  the  true  and  magnetic  meridians,  from 
which  the  angles  are  estimated,  are  different.  When  the  variation  is 
east,  the  magnetic  meridian  is  to  the  right  of  the  true  meridian  ;  when 


AZIMUTH  OF  A    TERRESTRIAL    OBJECT.  215 

Or,  numerically, 

Z  —  z  -j-  £,  when  the  azimuth  of  the  terrestrial  object  is 
greater  than  that  of  the  celestial ; 

Z  =  z  —  £,  when  it  is  less.  The  sign  of  £  should  be  noted  in 
the  observations. 

240.  2=NZ  M,  the  azimuth  of  the  celestial  body,  may  be 
found  from  an  observed  altitude  (Prob.  34),  or  from  the  local 
time  (Prob.  32).  In  the  first  case,  the  most  favorable  posi- 
tion is  on  or  nearest  the  prime  vertical ;  for  then  the  azimuth 
changes  most  slowly  with  the  altitude.  In  the  latter,  positions 
near  the  meridian  may  also  be  successfully  used. 

241.  £  =  M  Z  0,  the  azimuth  angle  between  the  two  ob- 
jects, may  be  found  in  one  of  the  following  ways  :  — 

1st  Method.     (By  direct  measurement.) 

M  Z  0,  being  a  horizontal  angle,  may  be  measured  directly 
by  a  theodolite  or  a  compass,  by  directing  the  line  of  sight 
of  the  instrument  first  to  one  of  the  objects  and  reading  the 
horizontal  circle,  then  to  the  other  and  reading  again.  The 
difference  of  the  two  readings  is  the  angle  required.  Or,  the 
telescope  or  sight-vanes  of  a  plane  table  may  be  directed  suc- 
cessively to  the  objects,  and  lines  drawn  upon  the  paper  along 
the  edge  of  the  ruler  in  its  two  positions,  and  the  angle  which 
they  form  measured  by  a  protracter. 

At  the  instant  when  the  observation  is  made  of  the  celestial 

the  variation  is  ivest,  the  magnetic  meridian  is  to  the  left  of  the  true 
meridian. 

It  is  necessary  to  distinguish  between  the  magnetic  bearing  and  the 
compass  bearing.  The  latter  is  affected  by  the  errors  of  the  instrument 
employed  and  by  local  disturbances  ;  the  former  is  free  from  them. 


216        *  NAVIGATION. 

object,  either  its  altitude  should  be  measured,  or  the  time  noted, 
so  as  to  find  its  azimuth  simultaneously. 

The  instrument  should  be  carefully  adjusted  and  levelled. 
With  the  compass  or  plane  table,  it  is  not  well  to  observe 
objects  whose  altitudes  are  greater  than  15°. 

A  theodolite  can  be  used  with  greater  precision  than  the 
other  instruments ;  but  the  greater  the  altitude  of  the  object, 
the  more  carefully  must  the  cross-threads  be  adjusted  to  the 
axis  of  collimation,  and  the  telescope  be  directed  to  the  object. 

The  error  of  collimation  is  eliminated  by  making  two  ob- 
servations with  the  telescope  reversed  either  in  is  Vs,  or  by 
rotation  on  its  axis.     Low  altitudes  are  generally  best. 

242.  If  the  sun  is  used,  each  limb  may  be  observed  alter- 
nately ;  or  a  separate  set  of  observations  may  be  made  for 
each. 

To  find  the  azimuth  reduction  for  semi-diameter,  when 
but  one  limb  is  observed ; 

Let  h  =  90°  —  Z  s  (Fig  51),  the  altitude  of 

the  sun, 

s  =  S  s,  its  semi-diameter, 

Z 
s'  =  S  Z  s,  the  reduction  of  the  azimuth 

for  the  semi-diameter. 

We  have  ~  „  sin  S  s 

sin  S  Z  s  = — — , 

sin  Z  s 

or,  since  s  and  it  are  small, 

s'  =  s  sec  A,  (151) 

which  is  the  reduction  required. 

The  sign  with  which  it  is  to  be  applied  depends  upon  the 
limb  observed. 


Fig.  43. 


AZIMUTH  OF  A    TERRESTRIAL   OBJECT.  217 

243.  If  the  observations  are  made  at  night,  and  the  ter- 
restrial object  is  invisible,  a  temporary  station  in  a  conve- 
nient position  may  be  used,  and  its  azimuth  found.  The 
horizontal  angle  between  this  and  the  terrestrial  object  may 
be  measured  by  daylight,  and  added  to,  or  subtracted  from, 
this  azimuth. 

A  board,  with  a  vertical  slit  and  a  light  behind  it,  forms 
a  convenient  mark  for  night  observations. 

The  place  of  the  theodolite  should  be  marked,  that  the 
instrument  may  be  replaced  in  the  same  position.  But  in 
doing  this,  and  selecting  the  temporary  station,  it  should  be 
kept  in  mind  that  a  change  of  the  position  of  the  instrument 
of  siVs  °f  the  distance  of  the  object  may  change  the  azimuth 
1' ;  or  of  2  o  oW(T  °f  the  distance  may  change  the  azimuth 
more  than  1". 

244.  2d  Method.  Finding  the  difference  of  azimuths  of 
a  celestial  and  a  terrestrial  object  by  a  sextant ;  sometimes 
called  an  "  astronomical  bearing." 

Measure  with  a  sextant  the  angular  distance  M  0  (Fig.  44) 
of  the  two  objects,  and  either  note  the  time  by  a  watch  reg- 
ulated to  local  time,  or  measure  simultaneously  the  altitude 
of  the  celestial  object.  Measure,  also,  the  altitude  of  the 
terrestrial  object  (if  it  is  not  in  the  horizon),  either  with  a 
theodolite  which  is  furnished  with  a  vertical  circle,  or  with 
a  sextant  above  the  water-line  at  the  base  of  the  object, 
when  there  is  one.  Correct  the  readings  of  the  instruments 
for  index  errors,  and  when  only  one  limb  of  the  sun  is  ob- 
served, for  semidiameter.* 

Observed  altitudes  of  either  object  above  the  water-line 

*  It  is  best  in  measuring  the  distance  of  the  sun  from  the  terrestrial 
object  to  use  each  limb  alternately. 


218 


NAVIGATION. 


are  also  to  be  corrected  for  the  dip  by  (33)  or  Table  14 
(Bo wd.),  if  the  horizon  is  free ;  but  by  (35)  or  Table  15 
(Bowd.),  if  the  horizon  is  obstructed. 
The  altitude  of  the  celestial  object, 
when  not  observed  simultaneously, 
may  be  interpolated  from  altitudes 
before  and  after,  by  means  of  the 
noted  times.  (Bowd.,  Art.  312.)  Or 
the  true  altitude  may  be  computed 
for  the  local  time  (Prob.  32  or  33) 
and  the  refraction  added  and  the 
parallax  subtracted  to  obtain  the  ap- 
parent altitude. 

Z  0  (Fig.  42),  the  apparent  altitude  of  0, 
JET  »  90°  -  Z  M,  the  apparent*  altitude  of  M.      - 
i)  =  MO,  the  corrected  distance. 


Let  N  =  90c 


We  have  then  in  the  triangle  M  Z  0  the  three  sides  from 
which  (  =  MZO  may  be  found  by  one  of  the  following 
formulas :  — 

1.  By  Sph.  Trig.  (164)  we  have 

un  *  *  -y  —  cos  Rr  co—r 

or,  letting  d  =  H'  —  ti,  "j 

sin  l  I  _  4  /fanj(l>  +  d)smi(J)-d)  \ 

2  4  "  V  COS  JSP  cos  h!  J       (152) 

2.  By  Sph.  Trig.  (165), 

1508  **-y-  cos  ir  COS  h' 


*  The  true  altitude  of  M  is  used  in  finding  z,  its  azimuth. 


AZIMUTH  OF  A    TERRESTRIAL   OBJECT.  219 

or,  putting 


s  =  i(ir  +  h'  +  D) 

,   j.         /cos  s  cos  (s  —  D) 
C°SH  =  V     cob  IT  cos  A' 


(153) 


(152)  is  preferable  when  £  <  90°  ;  (153),  when  £  >  90°. 

245.  If  0  is  in  the  true  horizon,  or  its  measured  altitude 
above  the  water  line  equals  the  dip,  h  =  0,  and  the  right 
triangle  M  m  0  gives 

cos  £  =  cos mO  =  cos  D  sec  IF ;  (154) 

or,  more  accurately,  when  £  is  small  (Sph.  Trig.,  105), 

tan  J  £  =  V  (tan  £(2>  +  ^0  tan  \  B  ~  ir)  •  (155) 

If  the  terrestrial  object  is  in  the  water-line,  h!  is  negative, 
and  equals  the  dip. 

246.  If  both  objects  are  in  the  horizon,  or  H  and  h  are 
equal  and  very  small,  we  have  simply 

£ '—■!>.  (156) 

In  general,  the  result  is  more  reliable  the  smaller  the  in- 
clination of  M  O  to  the  horizon.  If  M  O  is  perpendicular  to 
the  horizon,  the  problem  is  indeterminate  by  this  method. 

247.  If  the  terrestrial  object  presents  a  vertical  line  to 
which  the  sun's  disk  is  made  tangent,  the  reduction  of  the 
observed  distance  for  semi-diameter  is 

tf  =  s  sin  M  O  Z  (157) 

and  not  5,  the  semi-diameter  itself.  This  follows  from  the 
sun's  diameter  through  the  point  of  contact,  O,  being  per- 
pendicular to  the  vertical  circle  Z  O  and  not  in  the  direction 
of  the  distance  O  M. 


/^ 


220  NAVIGATION. 

As  the  altitude  of  the  terrestrial  object  is  always  very 
small,  we  may  find  M  0  Z  by  the  formula 

,^  ^  ~        sin  A' 
cos  M  0  Z  =  — : — =j 
sin  Ir 

jy  being  the  unreduced  distance. 

248.  When  precision  is  requisite,  the  axis  of  the  sextant 
with  which  the  angular  distance  is  measured  must  be  placed 
at  the  station  Z ;  and  if  the  object  seen  direct  is  sufficiently 
near,  the  parallactic  correction  must  be  added  to  the  sextant 
reading.     If 

A  represent  the  distance  of  the  object, 

dy  the  distance  of  the  axis  from  the  line  of  sight  or  axis  of 
the  telescope,  this  correction  is 

p  =  -^cosec  1"  =  206265"  —  (158) 

It  is  1',  when  A  =  3437.75  d. 

249.  If  the  distance  of  the  terrestrial  object  and  the  dif- 
ference of  level  above  or  below  the  level  of  the  instrument 
are  known,  we  may  find  its  angle  of  elevation,  nearly,  by  the 

formula  -^ 

tan  h'  =  — , 

A 

A  being  the  distance  of  the  object,  and 

E,  its  elevation  above  the  horizontal  plane  of  the  instrument. 

If  the  object  is  below  that  plane,  E  and  h!  will  have  the 
negative  sign. 

Note.  —  The  horizontal  angle  between  two  terrestrial  objects  may 
also  be  found  by  measuring  their  angular  distance  with  a  sextant,  and 
employing  the  same  formulas  (230  to  234)  as  for  a  celestial  and  terrestrial 
object ;  IT  and  h'  representing  their  apparent  angles  of  elevation.  Each 
of  these  may  be  found  by  direct  measurement,  or  from  the  known  dis- 
tance and  the  elevation,  or  depression,  from  the  horizontal  plane  of  the 


AZIMUTH  OF  A    TERRESTRIAL    OBJECT.  221 

observer.     If  the  two  objects  are  on  the  same  level  as  the  observer,  we 
have  simply  as  in  (234)  J  =  D. 

Example. 

1898,  May  16,  5h  45™  a.m.  in  lat.  38°  15'  K,  long.  76°  16' 
W. ;  the  angular  distance  of  the  sun's  centre  from  the  top  of 
a  light-house  measured  by  a  sextant  (O  to  the  right  of  L.  H.), 
75°  16'  25",  index  cor.  -  V  15" ;  altitude  of  O  above  the  sea- 
horizon  observed  at  the  same  time,  10°  18'  20",  index  cor.  -f 
2'  10" ;  observed  altitude  of  the  top  of  light-house  above  the 
water-line,  distant  7,300  feet,  1°  15'  20",  index  cor.,  +  2'  10"; 
height  of  eye,  20  feet ;  required  the  true  bearing  of  the  light- 
house. 
G.  m.  t.,  May  15,  22*  50m  04s  =  May  16  -  1M7 


Obs'dalt.Q            10°  18' 20" 

O's  dec,  May  16. 

I.  c.                             +2  10 

+  19°  10'  15".7     +  34".4 

Dip                             -  4  23 

r34".4 

Ap.  alt.     0            10   16  07 

-40  .2      j      3.4 

S.  D.                         +  15  51 

+  19   09  35  .5       I     2.4 

Ref.  and  Par.          —    5  03 

Ap.  alt.  -e,  H'=  10  31  58 

Obs'dalt.  L.  H.  1°  15' 20" 

Tr.  alt.   -e,  H  =  10   26  55 

I.C.                        +  2   10 

Ang.  dist.          =  75    15  10 

Dip                        -  9  56        by  (55) 

App.  alt.  L.  H.  1   07  34  =  h! 

Computation  by  (78)  and  (152). 

H=  10°  26' 55"  1.  sec  0.00726 

H'=  10°  31'  58"  1.  sec  0.00738 

L  =  38  15          1.  sec  0.10496 

h>  =    1   07  34    1.  sec  0.00008 

p  =  70  50  25 

d  =    9  24  24 

2  s  =119  32  20 

i  (D-M)   =  42    19  47     1.  sin  9.82827 

s  =  59  46  10    1.  cos  9.70198 

h  (D-d)   =  32   55  24     1.  sin  9.73521 

9.57094 

p—s  =  11   04  15     1.  cos  9.99184 

if  =37°  36'.2       1.  sin  9.78547 

9.80604 

J  =  75   12  .4 

i  Z  =  36°  53'          1.  cos  9.90302 

O's  Azimuth 

Z  =  N.  73  46  .0  E. 

True  bearing  of  L. 

H.  (Z-£)=N.    1°24'.4W. 

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